The document describes how to solve systems of equations by combining like terms. It explains that combining equations can speed up the process compared to graphing or substitution. An example problem demonstrates the steps: 1) choose a variable to eliminate, 2) make coefficients opposite and combine equations, 3) solve for the remaining variable, 4) plug back into the original equation to find the other variable, and 5) check the solution. The worked example finds the solution is x=2, y=2.
The substitution method is a technique for solving systems of linear equations. It involves solving one equation for one variable in terms of the other and substituting this expression into the second equation. The key steps are: 1) solve one equation for one variable, 2) substitute this expression into the other equation, 3) solve the resulting equation for the remaining variable, 4) check the solution by substituting back into the original equations. Examples demonstrate both consistent and inconsistent systems of equations solved using this method.
Lesson30 First Order Difference Equations HandoutMatthew Leingang
This document summarizes key concepts about first-order linear and nonlinear difference equations:
1) Linear difference equations of the form yk+1 = ayk + b can be solved explicitly using the formulas yk = ak(y0 - b/(1-a)) + b/(1-a).
2) For the logistic difference equation yk+1 = ryk(1-yk), the equilibria occur when r-1-ry=0 and are stable if the slope |g'(y)| = |r-2ry| is less than 1.
3) Both linear and nonlinear equations can be analyzed graphically using cobweb diagrams to determine stability of equilibria based
This document provides a lesson on solving multiplication equations. It begins with warm up problems involving division. It then presents an example problem and shows how to set up and solve a multiplication equation to find an unknown value. Additional examples demonstrate using multiplication equations to solve word problems involving area of rectangles. The document ensures understanding with practice problems and a lesson quiz.
The document is about difference equations and includes:
1) An introduction to difference equations, what they are, and their objectives.
2) Examples of testing solutions by plugging them into difference equations.
3) A "guess and check" method for finding the terms of a sequence defined by a difference equation.
1. The document presents a word problem about Diego's fruit stand. Diego distributed 12 boxes of fruit on the first day of April, 18 boxes on the second day, and so on with each day's amount increasing by 6 boxes.
2. To solve the problem, one can create a table listing the date, boxes distributed, and use the increasing pattern to determine the amounts. The expression an = 12 + 6(n-1) represents the number of boxes distributed on the nth day.
3. Using the expression, the number of boxes distributed on April 12th (78 boxes) and April 30th (186 boxes) can be calculated. The number of boxes to be distributed on May 20th (
This document contains an unsolved mathematics paper from 2007 with 42 multiple choice questions. The questions cover topics in algebra, trigonometry, calculus, vectors, and probability. The correct answers to each question are indicated by letters a, b, c, or d.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Lesson30 First Order Difference Equations SlidesMatthew Leingang
This document provides an overview of lesson 30 on first order difference equations. It includes:
1) Announcements about upcoming exams and homework due dates.
2) Instructions to make cobweb diagrams of various linear difference equations and a link to an online applet for doing so.
3) Facts and solutions about linear and nonlinear difference equations, including determining equilibria, stability conditions, and general solutions.
4) Practice problems involving guessing solutions, finding equilibria, and making conjectures about stability.
The substitution method is a technique for solving systems of linear equations. It involves solving one equation for one variable in terms of the other and substituting this expression into the second equation. The key steps are: 1) solve one equation for one variable, 2) substitute this expression into the other equation, 3) solve the resulting equation for the remaining variable, 4) check the solution by substituting back into the original equations. Examples demonstrate both consistent and inconsistent systems of equations solved using this method.
Lesson30 First Order Difference Equations HandoutMatthew Leingang
This document summarizes key concepts about first-order linear and nonlinear difference equations:
1) Linear difference equations of the form yk+1 = ayk + b can be solved explicitly using the formulas yk = ak(y0 - b/(1-a)) + b/(1-a).
2) For the logistic difference equation yk+1 = ryk(1-yk), the equilibria occur when r-1-ry=0 and are stable if the slope |g'(y)| = |r-2ry| is less than 1.
3) Both linear and nonlinear equations can be analyzed graphically using cobweb diagrams to determine stability of equilibria based
This document provides a lesson on solving multiplication equations. It begins with warm up problems involving division. It then presents an example problem and shows how to set up and solve a multiplication equation to find an unknown value. Additional examples demonstrate using multiplication equations to solve word problems involving area of rectangles. The document ensures understanding with practice problems and a lesson quiz.
The document is about difference equations and includes:
1) An introduction to difference equations, what they are, and their objectives.
2) Examples of testing solutions by plugging them into difference equations.
3) A "guess and check" method for finding the terms of a sequence defined by a difference equation.
1. The document presents a word problem about Diego's fruit stand. Diego distributed 12 boxes of fruit on the first day of April, 18 boxes on the second day, and so on with each day's amount increasing by 6 boxes.
2. To solve the problem, one can create a table listing the date, boxes distributed, and use the increasing pattern to determine the amounts. The expression an = 12 + 6(n-1) represents the number of boxes distributed on the nth day.
3. Using the expression, the number of boxes distributed on April 12th (78 boxes) and April 30th (186 boxes) can be calculated. The number of boxes to be distributed on May 20th (
This document contains an unsolved mathematics paper from 2007 with 42 multiple choice questions. The questions cover topics in algebra, trigonometry, calculus, vectors, and probability. The correct answers to each question are indicated by letters a, b, c, or d.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Lesson30 First Order Difference Equations SlidesMatthew Leingang
This document provides an overview of lesson 30 on first order difference equations. It includes:
1) Announcements about upcoming exams and homework due dates.
2) Instructions to make cobweb diagrams of various linear difference equations and a link to an online applet for doing so.
3) Facts and solutions about linear and nonlinear difference equations, including determining equilibria, stability conditions, and general solutions.
4) Practice problems involving guessing solutions, finding equilibria, and making conjectures about stability.
Lesson31 Higher Dimensional First Order Difference Equations SlidesMatthew Leingang
This document summarizes a lesson on higher dimensional difference equations. It discusses:
1) Linear systems described by equations of the form y(k+1) = Ay(k) and their solutions involving eigenvalues and eigenvectors of A.
2) Qualitative analysis of diagonal systems based on the magnitudes of the eigenvalues determining behaviors like attraction, repulsion, or saddle points.
3) The nonlinear case where equilibria y* are found as solutions to g(y*)=y* and stability is determined by eigenvalues of the Jacobian matrix Dg(y*) evaluated at the equilibria.
Lesson32 Second Order Difference Equations SlidesMatthew Leingang
This document discusses solving second-order difference equations by treating them as systems of first-order equations. It provides the example of the Fibonacci sequence, defined by f(k+2)=f(k)+f(k+1), and shows how to set it up as the system y(k+1)=Ay(k) where y(k)=[f(k) g(k)]^T. The eigenvalues and eigenvectors of the matrix A are found, allowing the solution to be written as a combination of terms involving the eigenvalues. The constants are then determined using the initial conditions to obtain the final solution for f(k).
1. The document provides instructions and examples for solving systems of equations using elimination.
2. It explains that you may need to multiply one or both equations by a number to get opposite terms that can be eliminated. Then you add the equations to solve for one variable, substitute to solve for the other, and write the solution as an ordered pair.
3. Several examples of systems of equations are given and students are instructed to use elimination to solve them.
1. The passage provides an unsolved mathematics exam from 2005 containing 32 multiple choice problems related to topics like probability, geometry, trigonometry, and calculus.
2. The problems cover a wide range of mathematical concepts tested through multiple choice questions with 4 answer options for each problem.
3. No solutions or answers are provided, as the exam paper is labeled as "unsolved".
The document is a solutions manual providing answers to even-numbered exercises from the textbook "A First Look at Rigorous Probability Theory" by Jeffrey S. Rosenthal. It was compiled by Mohsen Soltanifar, Longhai Li, and Jeffrey S. Rosenthal, and published by World Scientific Publishing Co. in 2010 under a Creative Commons license. The preface explains that providing solutions to only the even exercises aims to help students while still allowing instructors to assign odd exercises for coursework. It notes that errors may be present and should be reported.
The document provides data and questions about 6 topics: body mass of children, math test marks, ages of golf club members, charity donations, student masses, and student pocket money. For each topic, tables are completed with frequency distributions, measures of center are calculated (mean, median, mode), and graphs (histogram, frequency polygon, ogive) are drawn to represent the data. The answers and calculations are provided in a detailed manner across multiple pages.
This document is a teacher's lesson plan for teaching students how to solve one-step equations. It includes learning objectives, examples of solving equations step-by-step with explanations, think-pair-share activities where students solve equations individually and then discuss with partners, and homework assignments of word problems requiring students to set up and solve equations. The lesson emphasizes adding or subtracting values from both sides of an equation to isolate the variable.
04 structured prediction and energy minimization part 1zukun
The document discusses structured prediction problems and energy minimization approaches. It describes how structured prediction involves finding the optimal prediction y* from a set of possibilities Y that maximizes an objective function g(x,y). Exactly solving such problems is difficult because Y is large but finite. The document outlines desirable properties for algorithms that evaluate the prediction function f(x), including being general, optimal, efficient, having integral solutions, and being deterministic. However, achieving all properties simultaneously is impossible for hard problems. Approaches give up certain properties, like generality, to design algorithms that satisfy the remaining desirable properties.
The document contains 30 multiple choice questions from a past UPSEE mathematics exam. The questions cover a range of topics including: [1] calculating time taken to cross a canal based on speed and direction of flow; [2] determining the derivative of an exponential function; [3] finding the velocity of a particle with given acceleration over time.] The full document provides the questions and multiple choice answers but no solutions.
This document contains notes from a Calculus I class at New York University. It discusses related rates problems, which involve taking derivatives of equations relating changing quantities to determine rates of change. The document provides examples of related rates problems involving an oil slick, two people walking towards and away from each other, and electrical resistors. It also outlines strategies for solving related rates problems, such as drawing diagrams, introducing notation, relating quantities with equations, and using the chain rule to solve for unknown rates.
The document provides examples of graphing quadratic equations by making tables of values and plotting points. Example 1 graphs the equation y = x^2 + 2 by making a table with x-values from -2 to 2 and the corresponding y-values. The points are then plotted and connected to show the parabolic shape. Example 2 graphs y = 2x^2 + 3x - 7 by the same process and states that the domain is all real numbers and the range varies between approximately -8 and 20.
The document provides definitions and theorems about secants, tangents, and angle measures formed by lines that intersect within or outside of a circle. It includes examples that apply the theorems to find measures of angles and arc lengths. Theorems relate the measures of angles formed to the measures of intercepted arcs when lines intersect inside or outside the circle. Examples work through applications of the theorems to find specific angle measures.
This document contains a step-by-step example problem that determines the value of k in various mathematical functions where the point (10,20) is given. For each function, it also calculates the corresponding y-value when x is 30. The key steps are:
1) Determine k by plugging the given point into each function
2) Plug x=30 into each function to determine the corresponding y-values
3) The full example problem solves for k and y-values for 4 different functions: y=kx, y=k/x, y=kx^2, y=k/x^2
The document discusses how to write and graph linear inequalities in two variables. It begins by defining key vocabulary terms related to linear inequalities such as open and closed half-planes, boundary lines, and test points. It then provides examples of how to determine if a point satisfies a linear inequality by testing it, and how to graph linear inequalities by plotting the boundary line and shading the appropriate half-plane based on the inequality symbols. Examples show both solid and dashed boundary lines being graphed for various inequalities.
This document provides instruction on how to multiply monomials by using the rules of exponents. It begins with an essential question about how to multiply monomials using exponents. It then explains the rules: like terms are multiplied, numbers and variables are multiplied separately. Examples are provided to demonstrate multiplying monomials like -2xy, 3xy(-4xy)^3, and 7x(7x). The document concludes with a word problem involving multiplying monomials to calculate dividend earnings from stock shares.
The document provides information about direct variation and direct square variation functions, including definitions of key terms like constant of variation. It gives examples of how to set up and solve direct variation and direct square variation word problems. The first example involves using direct variation to determine the volume of a gas at a given temperature. The second example uses direct square variation to calculate the stopping distance of a car traveling at a certain speed.
1) The document discusses rational power functions and provides the standard frequencies of musical notes starting from A above middle C.
2) It gives an expression for calculating the frequency of any note n notes above A as 440*(2^(1/12))^n.
3) Using this expression, it calculates the frequency of the G note that is two notes below A.
The document provides information about classifying quadrilaterals and properties of parallelograms. It defines key terms related to quadrilaterals and parallelograms. It presents a hierarchy of quadrilaterals from most general to most specific. Properties of parallelograms are outlined, including that opposite sides are congruent, opposite angles are congruent, and diagonals bisect each other. An example problem demonstrates using properties of parallelograms to solve for lengths of sides and diagonals.
This document provides examples for converting between customary and metric units of length, capacity, and weight. It begins with conversion factors for units of length like inches to centimeters, yards to meters, and miles to kilometers. Examples are given to convert specific values like 26 centimeters to inches. The document then provides conversion factors and examples for units of capacity, like quarts to liters and pints to liters. Finally, conversion factors and examples are given for units of weight and mass, such as pounds to kilograms and grams to ounces. Step-by-step workings are shown for each example conversion.
This document summarizes key concepts about isosceles and equilateral triangles. It defines important vocabulary like the legs and vertex angle of an isosceles triangle. It states theorems like if two sides of a triangle are congruent, the angles opposite them are congruent. It also defines properties of equilateral triangles, like they are equiangular and each angle is 60 degrees. Examples demonstrate using these concepts to find missing angle and side measures. The document concludes with assigning practice problems.
The document discusses proving angle relationships through postulates and theorems. It introduces the protractor postulate, angle addition postulate, and theorems regarding supplementary, complementary, congruent, and right angles. Examples are provided to demonstrate using these concepts to prove and determine angle measures.
Lesson31 Higher Dimensional First Order Difference Equations SlidesMatthew Leingang
This document summarizes a lesson on higher dimensional difference equations. It discusses:
1) Linear systems described by equations of the form y(k+1) = Ay(k) and their solutions involving eigenvalues and eigenvectors of A.
2) Qualitative analysis of diagonal systems based on the magnitudes of the eigenvalues determining behaviors like attraction, repulsion, or saddle points.
3) The nonlinear case where equilibria y* are found as solutions to g(y*)=y* and stability is determined by eigenvalues of the Jacobian matrix Dg(y*) evaluated at the equilibria.
Lesson32 Second Order Difference Equations SlidesMatthew Leingang
This document discusses solving second-order difference equations by treating them as systems of first-order equations. It provides the example of the Fibonacci sequence, defined by f(k+2)=f(k)+f(k+1), and shows how to set it up as the system y(k+1)=Ay(k) where y(k)=[f(k) g(k)]^T. The eigenvalues and eigenvectors of the matrix A are found, allowing the solution to be written as a combination of terms involving the eigenvalues. The constants are then determined using the initial conditions to obtain the final solution for f(k).
1. The document provides instructions and examples for solving systems of equations using elimination.
2. It explains that you may need to multiply one or both equations by a number to get opposite terms that can be eliminated. Then you add the equations to solve for one variable, substitute to solve for the other, and write the solution as an ordered pair.
3. Several examples of systems of equations are given and students are instructed to use elimination to solve them.
1. The passage provides an unsolved mathematics exam from 2005 containing 32 multiple choice problems related to topics like probability, geometry, trigonometry, and calculus.
2. The problems cover a wide range of mathematical concepts tested through multiple choice questions with 4 answer options for each problem.
3. No solutions or answers are provided, as the exam paper is labeled as "unsolved".
The document is a solutions manual providing answers to even-numbered exercises from the textbook "A First Look at Rigorous Probability Theory" by Jeffrey S. Rosenthal. It was compiled by Mohsen Soltanifar, Longhai Li, and Jeffrey S. Rosenthal, and published by World Scientific Publishing Co. in 2010 under a Creative Commons license. The preface explains that providing solutions to only the even exercises aims to help students while still allowing instructors to assign odd exercises for coursework. It notes that errors may be present and should be reported.
The document provides data and questions about 6 topics: body mass of children, math test marks, ages of golf club members, charity donations, student masses, and student pocket money. For each topic, tables are completed with frequency distributions, measures of center are calculated (mean, median, mode), and graphs (histogram, frequency polygon, ogive) are drawn to represent the data. The answers and calculations are provided in a detailed manner across multiple pages.
This document is a teacher's lesson plan for teaching students how to solve one-step equations. It includes learning objectives, examples of solving equations step-by-step with explanations, think-pair-share activities where students solve equations individually and then discuss with partners, and homework assignments of word problems requiring students to set up and solve equations. The lesson emphasizes adding or subtracting values from both sides of an equation to isolate the variable.
04 structured prediction and energy minimization part 1zukun
The document discusses structured prediction problems and energy minimization approaches. It describes how structured prediction involves finding the optimal prediction y* from a set of possibilities Y that maximizes an objective function g(x,y). Exactly solving such problems is difficult because Y is large but finite. The document outlines desirable properties for algorithms that evaluate the prediction function f(x), including being general, optimal, efficient, having integral solutions, and being deterministic. However, achieving all properties simultaneously is impossible for hard problems. Approaches give up certain properties, like generality, to design algorithms that satisfy the remaining desirable properties.
The document contains 30 multiple choice questions from a past UPSEE mathematics exam. The questions cover a range of topics including: [1] calculating time taken to cross a canal based on speed and direction of flow; [2] determining the derivative of an exponential function; [3] finding the velocity of a particle with given acceleration over time.] The full document provides the questions and multiple choice answers but no solutions.
This document contains notes from a Calculus I class at New York University. It discusses related rates problems, which involve taking derivatives of equations relating changing quantities to determine rates of change. The document provides examples of related rates problems involving an oil slick, two people walking towards and away from each other, and electrical resistors. It also outlines strategies for solving related rates problems, such as drawing diagrams, introducing notation, relating quantities with equations, and using the chain rule to solve for unknown rates.
The document provides examples of graphing quadratic equations by making tables of values and plotting points. Example 1 graphs the equation y = x^2 + 2 by making a table with x-values from -2 to 2 and the corresponding y-values. The points are then plotted and connected to show the parabolic shape. Example 2 graphs y = 2x^2 + 3x - 7 by the same process and states that the domain is all real numbers and the range varies between approximately -8 and 20.
The document provides definitions and theorems about secants, tangents, and angle measures formed by lines that intersect within or outside of a circle. It includes examples that apply the theorems to find measures of angles and arc lengths. Theorems relate the measures of angles formed to the measures of intercepted arcs when lines intersect inside or outside the circle. Examples work through applications of the theorems to find specific angle measures.
This document contains a step-by-step example problem that determines the value of k in various mathematical functions where the point (10,20) is given. For each function, it also calculates the corresponding y-value when x is 30. The key steps are:
1) Determine k by plugging the given point into each function
2) Plug x=30 into each function to determine the corresponding y-values
3) The full example problem solves for k and y-values for 4 different functions: y=kx, y=k/x, y=kx^2, y=k/x^2
The document discusses how to write and graph linear inequalities in two variables. It begins by defining key vocabulary terms related to linear inequalities such as open and closed half-planes, boundary lines, and test points. It then provides examples of how to determine if a point satisfies a linear inequality by testing it, and how to graph linear inequalities by plotting the boundary line and shading the appropriate half-plane based on the inequality symbols. Examples show both solid and dashed boundary lines being graphed for various inequalities.
This document provides instruction on how to multiply monomials by using the rules of exponents. It begins with an essential question about how to multiply monomials using exponents. It then explains the rules: like terms are multiplied, numbers and variables are multiplied separately. Examples are provided to demonstrate multiplying monomials like -2xy, 3xy(-4xy)^3, and 7x(7x). The document concludes with a word problem involving multiplying monomials to calculate dividend earnings from stock shares.
The document provides information about direct variation and direct square variation functions, including definitions of key terms like constant of variation. It gives examples of how to set up and solve direct variation and direct square variation word problems. The first example involves using direct variation to determine the volume of a gas at a given temperature. The second example uses direct square variation to calculate the stopping distance of a car traveling at a certain speed.
1) The document discusses rational power functions and provides the standard frequencies of musical notes starting from A above middle C.
2) It gives an expression for calculating the frequency of any note n notes above A as 440*(2^(1/12))^n.
3) Using this expression, it calculates the frequency of the G note that is two notes below A.
The document provides information about classifying quadrilaterals and properties of parallelograms. It defines key terms related to quadrilaterals and parallelograms. It presents a hierarchy of quadrilaterals from most general to most specific. Properties of parallelograms are outlined, including that opposite sides are congruent, opposite angles are congruent, and diagonals bisect each other. An example problem demonstrates using properties of parallelograms to solve for lengths of sides and diagonals.
This document provides examples for converting between customary and metric units of length, capacity, and weight. It begins with conversion factors for units of length like inches to centimeters, yards to meters, and miles to kilometers. Examples are given to convert specific values like 26 centimeters to inches. The document then provides conversion factors and examples for units of capacity, like quarts to liters and pints to liters. Finally, conversion factors and examples are given for units of weight and mass, such as pounds to kilograms and grams to ounces. Step-by-step workings are shown for each example conversion.
This document summarizes key concepts about isosceles and equilateral triangles. It defines important vocabulary like the legs and vertex angle of an isosceles triangle. It states theorems like if two sides of a triangle are congruent, the angles opposite them are congruent. It also defines properties of equilateral triangles, like they are equiangular and each angle is 60 degrees. Examples demonstrate using these concepts to find missing angle and side measures. The document concludes with assigning practice problems.
The document discusses proving angle relationships through postulates and theorems. It introduces the protractor postulate, angle addition postulate, and theorems regarding supplementary, complementary, congruent, and right angles. Examples are provided to demonstrate using these concepts to prove and determine angle measures.
This document provides an overview of deductive reasoning, including the Law of Detachment and Law of Syllogism. It defines deductive reasoning as using facts, rules, and definitions to reach a logical conclusion. The Law of Detachment states that if p implies q is true and p is true, then q must be true. The Law of Syllogism states that if p implies q and q implies r are both true, then p implies r must be true. The document provides examples and problems applying these laws of deductive reasoning.
The document provides information on algebraic proofs and two-column proofs. It defines an algebraic proof as using a series of algebraic steps to solve problems and justify steps. A two-column proof is defined as having one column for statements and a second column for justifying each statement. Properties of equality like addition, subtraction, multiplication, and division properties are also defined for writing algebraic proofs. An example problem is worked through step-by-step to demonstrate an algebraic proof.
This document introduces key concepts in geometry including points, lines, planes, collinear points, coplanar points, and intersections. It defines points as having no size or shape, lines as infinite sets of points with no thickness, and planes as flat surfaces determined by three or more points that extend infinitely. Examples demonstrate identifying geometric objects from diagrams and real-world situations. Vocabulary and concepts are applied in problems identifying and relating points, lines and planes.
The document defines and provides examples of different types of angle relationships including adjacent angles, linear pairs, vertical angles, complementary angles, and supplementary angles. It also gives examples of using these concepts to find missing angle measures. For example, if two angles are vertical angles and one is measured as 72 degrees, then the other is also 72 degrees. It also shows how to set up and solve equations to determine missing angle measures or find values that satisfy given conditions, such as finding x and y values so that two lines are perpendicular.
The document discusses medians and altitudes of triangles. It defines key terms like median, centroid, altitude and orthocenter. A median connects a vertex to the midpoint of the opposite side, while an altitude connects a vertex perpendicular to the opposite side. The centroid is the point where the medians intersect and is always inside the triangle. The orthocenter is the point where the altitudes intersect. Examples are provided to demonstrate calculating lengths and finding points related to medians, altitudes, centroids and orthocenters using given information about triangles.
This document defines key terms related to congruent triangles such as congruent, congruent polygons, and corresponding parts. It also defines the Third Angle Theorem. Examples are provided to demonstrate how to prove triangles are congruent by identifying corresponding congruent parts and writing congruence statements. The document also includes multi-step proofs involving congruent triangles.
The document discusses writing equations of lines. It provides two forms for writing equations of lines: slope-intercept form (y=mx+b) and point-slope form (y-y1=m(x-x1)). An example problem finds the slope and y-intercept of two lines given by their equations in slope-intercept form and graphs the lines.
The document discusses solving systems of equations by combining equations through addition or multiplication. It explains that combining equations can speed up the process of solving systems compared to graphing or substitution. An example problem demonstrates the steps: 1) choose a variable to eliminate, 2) make the coefficients opposite to combine equations, 3) solve the combined equation for one variable, 4) substitute back into the original equation to find the other variable. Checking the solution verifies the method works.
The document contains examples and explanations of solving systems of equations by substitution. In Example 1, a system with two equations and two variables is solved to find the solution (2,4). In Example 2, a real-world word problem is modeled with a system of three equations with three variables to represent the number of different types of tickets printed for a play. The system is solved to find the numbers of adult (A=500), student (S=1000), and children's (C=250) tickets printed.
Steps are shown for solving systems of equations by the elimination method, including multiplying one equation by a constant to eliminate a variable before adding the equations. Examples are provided of solving systems by substitution, where one variable is solved for in one equation and substituted into the other equation. Practice problems are given for students to apply
The document provides examples for solving systems of equations by substitution. It includes worked examples showing the step-by-step process of solving systems with 2-3 equations by substituting values from one equation into another and solving for the variables. The examples determine the number of different types of tickets (adult, student, children's) printed for a school play given the total number of tickets and relationships between the ticket amounts.
This document contains class notes and materials on solving various types of algebraic equations. It discusses tips for solving literal equations, such as distributing terms when a variable is inside parentheses. It also explains the three possible outcomes when solving equations with variables on both sides: a single number solution, no solution, or an identity. The document provides examples of solving proportions and fractional equations. It concludes with examples of problems students most missed on a post-test related to rocket burning times, equations with fractions, and finding the largest of three consecutive even numbers.
The document discusses solving equations, including equations with unknowns on both sides and with brackets. It provides examples of solving various types of equations, such as equations with fractions or variables on both sides. Strategies for solving equations include collecting like terms, using the inverse operation to isolate the variable, and expanding any brackets before solving.
This document discusses different types of equations including rational equations, work rate problems, equations with radicals, and equations that are quadratic in form. It provides examples of solving each type of equation and emphasizes the importance of checking proposed solutions. Key steps include multiplying both sides by a common denominator for rational equations, setting up and solving work rate equations, isolating and eliminating radicals through exponentiation, and transforming equations into quadratic form.
The document discusses solving equations by using inverse operations such as addition and subtraction to isolate the variable. It provides examples of solving one-step and multi-step equations, as well as using proportions to solve word problems involving rates, conversions, and recipes. Key terms discussed include coefficients, constants, variables, and inverse operations.
1) The document provides instructions for an assignment that is due on Wednesday January 9th and asks if a test was signed.
2) It then gives examples of graphing linear equations and explains the substitution method for solving systems of linear equations.
3) Several examples using the substitution method to solve systems of linear equations are shown. The final example asks to find two numbers where the sum is 180 and the difference is 24.
This document provides examples of solving systems of linear equations by elimination. It explains the steps:
1) Write the system so that like terms are aligned.
2) Eliminate one variable by adding or subtracting the equations.
3) Substitute the value into one equation to solve for the other variable.
4) Write the solution as an ordered pair and check.
It shows how to multiply equations by a number to produce opposite coefficients for elimination. Examples demonstrate solving systems by addition, subtraction, and multiplication.
The document provides examples of solving different types of linear systems of equations, including graphing, substitution, and addition methods. It demonstrates setting up and solving systems to find unknown variables from word problems involving gardeners and helpers earning different amounts. The final section defines a linear system as a set of two or more linear equations.
Math 8-Lessslayyyyyyyyurrrrrrrrrron 1.pdfaflores17
This document provides an overview of multiplying polynomials. It defines factors and products, and provides examples of multiplying monomials using exponent rules. It also covers special cases like perfect square trinomials, difference of squares, sum and difference of cubes, and provides practice problems for the reader.
1) The document defines terms, constants, coefficients, and discusses how to solve one-step and two-step equations. It provides examples of solving equations involving addition, subtraction, multiplication and division.
2) Sample one-step equations are provided along with the steps to solve each type. Equations involving addition, subtraction, multiplication and division are worked through as examples.
3) A two-step practice problem is given along with the answers. Solving two-step equations is introduced.
Q1 week 1 (common monomial,sum & diffrence of two cubes,difference of tw...Walden Macabuhay
It consists of ten units in which the first unit focuses on the special products and factors. Its deals with the study of rational algebraic expressions. It aims to empower students with life – long learning and helps them to attain functional literacy. The call of the K to 12 curriculum allow the students to have an active involvement in learning through demonstration of skills, manifestations of communication skills, development of analytical and creative thinking and understanding of mathematical applications and connections.
The document provides examples for solving one-step, two-step, and multi-step equations. It begins with warm-up problems and then works through examples of solving equations with variables on both sides, combining like terms, and clearing fractions by multiplying both sides by the least common denominator. The examples are accompanied by step-by-step explanations and checks of the solutions. Additional practice problems and lessons on solving equations are also included.
1. The document discusses factors that affect the world's environment like population growth and provides population data from 1950 to 2050 in 10-year intervals.
2. It asks which 10-year period saw the largest increase in population and which saw the largest percentage increase.
3. The chapter aims to refresh skills working with numbers expressed as fractions, decimals, percentages, and indexes and applying them to real-life situations.
The document provides an overview of key topics in quadratic equations, including solving quadratic equations by factorizing, completing the square, and using the quadratic formula. It discusses why quadratics are important, such as in modeling projectile motion or summations, and provides examples of solving quadratic equations and completing the square to put them in standard form. The document also includes interactive tests and exercises to help students practice these skills in working with quadratic equations.
The document discusses power series representations of functions. Power series allow functions that cannot be integrated, like ex2, to be approximated by polynomials which can be integrated. This allows integrals to be approximated to any degree of accuracy. Power series also allow irrational numbers like π to be expressed as an infinite decimal expansion. Power series approximations can be used to find approximate solutions to difficult differential equations.
To solve trigonometric equations:
1) Use standard algebraic techniques such as collecting like terms and factoring to isolate the trigonometric function.
2) Many equations are of quadratic type (ax^2 + bx + c = 0) and can be solved by factoring or using the Quadratic Formula.
3) Equations involving multiple angles (sin ku, cos ku) are first solved for ku, then divided by k.
4) Inverse trigonometric functions can be used to solve equations, with the domain restrictions of the inverse functions considered.
The document defines and provides examples of angle relationships including adjacent angles, linear pairs, vertical angles, complementary angles, and supplementary angles. It defines each term and provides examples of identifying angle pairs that satisfy each relationship. It also includes examples of using properties of these angle relationships to solve problems, such as finding missing angle measures.
The document defines various terms related to angle measure including ray, angle, vertex, acute angle, obtuse angle, and angle bisector. It then provides examples measuring angles in a figure and solving an equation involving angle measures.
This document discusses finding points and midpoints on line segments. It defines midpoint as the point halfway between two endpoints and provides the formula to calculate it. Several examples are given to demonstrate how to find the midpoint of a segment, locate a point at a fractional distance from one endpoint, and find a point where the ratio of distances from the endpoints is a given ratio. The key concepts covered are calculating midpoints using averages of x- and y-coordinates, and setting up and solving equations to locate interior points using fractional or ratio distances along a segment.
This document defines key vocabulary terms related to line segments and distance. It defines a line segment as a portion of a line distinguished by endpoints, and defines betweenness of points and the term "between." It also defines congruent segments, constructions, distance, and irrational numbers. Several examples are provided to demonstrate calculating distances between points on a number line, using a ruler to measure segments, and applying the Pythagorean theorem.
This document defines key vocabulary terms related to line segments and distance, including line segment, betweenness of points, congruent segments, and distance formula. It provides examples of calculating distances between points on number lines and using the Pythagorean theorem to find distances between points graphed on a coordinate plane. Examples include measuring line segments with a ruler, finding distances by adding or subtracting measures, and applying the distance formula and Pythagorean theorem to solve for unknown distances.
This document introduces basic geometry concepts such as points, lines, planes, and their intersections. It defines a point as having no size or shape, a line as an infinite set of collinear points, and a plane as a flat surface extending indefinitely. Examples demonstrate identifying geometric shapes from real-world objects and graphing points and lines on a coordinate plane. The summary defines key terms and provides examples of geometric concepts and relationships.
The document discusses inverse functions and relations. It defines an inverse relation as one where the coordinates of a relation are switched, and an inverse function as one where the domain and range of a function are switched. It provides examples of finding the inverse of specific relations and functions by switching their coordinates or domain and range. It also discusses how to determine if two functions are inverses using their graphs and the horizontal line test.
The document discusses composition of functions. It defines composition of functions as using the output of one function as the input of another. It provides an example of composing two functions f and g, showing the steps of evaluating f(g(x)) and g(f(x)) at different values of x. Another example is given with two functions defined by sets of ordered pairs, finding the compositions f∘g and g∘f by evaluating them at different inputs and stating their domains and ranges.
The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, and division. Definitions of each operation are provided, along with examples of applying the operations to specific functions. Addition of functions involves adding the outputs of each function, subtraction involves subtracting the outputs, multiplication involves multiplying the outputs, and division involves dividing the outputs given the denominator function is not equal to 0. Several examples are worked through applying the different operations to functions like f(x)=2x and g(x)=-x+5. The examples also demonstrate evaluating composite functions and restricting domains as needed.
The document discusses determining the number and type of roots of polynomial equations. It states that every polynomial with degree greater than zero has at least one root in the set of complex numbers according to the Fundamental Theorem of Algebra. Descartes' Rule of Signs is introduced, which relates the number of changes in sign of a polynomial's terms to its possible positive and negative real roots. An example problem is worked through applying these concepts to determine the possible number of positive, negative, and imaginary roots.
This document discusses synthetic division and the remainder and factor theorems. It provides examples of using synthetic division to evaluate functions, determine the number of terms in a sequence, and factor polynomials. The key steps of synthetic division are shown, along with checking the remainder and determining common factors. Three examples are worked through to demonstrate these concepts.
This document discusses solving polynomial equations by factoring polynomials. It begins with essential questions and vocabulary about factoring polynomials and solving polynomial equations by factoring. It then provides the number of terms in a polynomial and the corresponding factoring technique that can be used. Examples of factoring various polynomials are also provided. The document aims to teach students how to factor polynomials and solve polynomial equations by factoring.
The document defines key terms and theorems related to trapezoids and kites. It provides definitions for trapezoid, bases, legs of a trapezoid, base angles, isosceles trapezoid, midsegment of a trapezoid, and kite. It also lists theorems about properties of isosceles trapezoids and kites. Two examples problems are included, one finding measures of an isosceles trapezoid and another showing a quadrilateral is a trapezoid.
The document discusses rhombi and squares. It defines a rhombus as a parallelogram with four congruent sides and gives its properties. A square is defined as a parallelogram with four right angles and four congruent sides. The document provides theorems for identifying rhombi and squares. It then gives examples of using the properties and theorems to determine if a shape is a rhombus, rectangle, or square.
The document discusses properties of rectangles. A rectangle is defined as a parallelogram with four right angles. The key properties are that opposite sides are parallel and congruent, opposite angles are congruent, and consecutive angles are supplementary. The diagonals of a rectangle bisect each other and are congruent. Theorems are presented regarding the diagonals of rectangles. Examples apply the properties of rectangles to find missing side lengths, angles, and diagonals. One example uses the distance formula and slope to determine if a quadrilateral is a rectangle.
The document discusses properties of parallelograms and provides examples of determining if a quadrilateral is a parallelogram. It defines four theorems for identifying parallelograms based on opposite sides, opposite angles, bisecting diagonals, and parallel/congruent sides. Examples solve systems of equations to find values of variables such that the quadrilaterals satisfy parallelogram properties. One example uses slopes of side segments to show a quadrilateral is a parallelogram due to parallel opposite sides.
The document discusses properties of parallelograms. It defines a parallelogram as a quadrilateral with two pairs of parallel sides. It then lists several properties of parallelograms: opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and if one angle is a right angle all angles are right angles. It also discusses properties of diagonals in parallelograms, including that diagonals bisect each other and divide the parallelogram into two congruent triangles. Several examples demonstrate using these properties to solve problems about parallelograms.
The document summarizes key concepts about polygons, including:
- The sum of the interior angles of a polygon with n sides is (n-2)180 degrees.
- The sum of the exterior angles of a polygon is 360 degrees.
- Examples are provided to demonstrate calculating sums of interior/exterior angles and finding missing angle measures using angle sums.
- Regular polygons are defined by their number of sides.
The document discusses analyzing graphs of polynomial functions. It provides examples of locating real zeros of polynomials using the location principle and estimating relative maxima and minima. Example 1 analyzes the polynomial f(x) = x^4 - x^3 - 4x^2 + 1 and locates its real zeros between consecutive integer values. Example 2 graphs the polynomial f(x) = x^3 - 3x^2 + 5 and estimates the x-coordinates of relative maxima and minima.
This document discusses polynomial functions. It defines key terms like polynomial in one variable, leading coefficient, and polynomial function. It provides examples of power functions of varying degrees like quadratic, cubic, quartic and quintic functions. The document also includes examples of evaluating polynomial functions, finding degrees and leading coefficients, graphing polynomial functions from tables of values, and describing properties of graphs.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Walmart Business+ and Spark Good for Nonprofits.pdf
Int Math 2 Section 8-4 1011
1. Section 8-4
Solve Systems by Adding, Subtracting, and
Multiplying (Linear Combinations)
Sun, Apr 10
2. Essential Questions
How do you solve systems of equations
by adding and subtracting?
How do you solve systems of equations
by adding and multiplying?
Where you’ll see this:
Landscaping, construction, sports
Sun, Apr 10
5. Why another method?
Graphing: Problems with accuracy
Substitution: Need an isolated variable
Sun, Apr 10
6. Why another method?
Graphing: Problems with accuracy
Substitution: Need an isolated variable
Combinations: Could speed up
process
Sun, Apr 10
7. Why another method?
Graphing: Problems with accuracy
Substitution: Need an isolated variable
Combinations: Could speed up
process
Look at the system to determine best method
Sun, Apr 10
10. Solve by Combinations
1. Choose a variable to eliminate (your choice).
2. Make the coefficients of that variable
opposite. You might need to multiply to do
this. Then combine equations.
Sun, Apr 10
11. Solve by Combinations
1. Choose a variable to eliminate (your choice).
2. Make the coefficients of that variable
opposite. You might need to multiply to do
this. Then combine equations.
3. Solve for the remaining variable.
Sun, Apr 10
12. Solve by Combinations
1. Choose a variable to eliminate (your choice).
2. Make the coefficients of that variable
opposite. You might need to multiply to do
this. Then combine equations.
3. Solve for the remaining variable.
4. Plug back into an original equation to find
the other variable.
Sun, Apr 10
13. Solve by Combinations
1. Choose a variable to eliminate (your choice).
2. Make the coefficients of that variable
opposite. You might need to multiply to do
this. Then combine equations.
3. Solve for the remaining variable.
4. Plug back into an original equation to find
the other variable.
5. Check and rewrite the answer.
Sun, Apr 10
14. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
Sun, Apr 10
15. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
Sun, Apr 10
16. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
2x + 7 y = 18
Sun, Apr 10
17. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
2x + 7 y = 18
Sun, Apr 10
18. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
2x + 7 y = 18
10 y = 20
Sun, Apr 10
19. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
2x + 7 y = 18
10 y = 20
10 10
Sun, Apr 10
20. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
2x + 7 y = 18
10 y = 20
10 10
y=2
Sun, Apr 10
21. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
10 y = 20
10 10
y=2
Sun, Apr 10
22. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20
10 10
y=2
Sun, Apr 10
23. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10
y=2
Sun, Apr 10
24. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2
Sun, Apr 10
25. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2
Sun, Apr 10
26. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩
−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2
x=2
Sun, Apr 10
27. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩ Check:
−2x + 3 y = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2
x=2
Sun, Apr 10
28. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩ Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2
x=2
Sun, Apr 10
29. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩ Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18 −4 + 6 = 2
−2x + 6 = 2
10 y = 20 −6 −6
10 10 −2x = −4
y=2 −2 −2
x=2
Sun, Apr 10
30. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩ Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18 −4 + 6 = 2
−2x + 6 = 2
10 y = 20 −6 −6 2(2) + 7(2) = 18
10 10 −2x = −4
y=2 −2 −2
x=2
Sun, Apr 10
31. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩ Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18 −4 + 6 = 2
−2x + 6 = 2
10 y = 20 −6 −6 2(2) + 7(2) = 18
10 10 −2x = −4 4 + 14 = 18
y=2 −2 −2
x=2
Sun, Apr 10
32. Example 1
Solve by combining the equations
⎧−2x + 3 y = 2
⎪
a. ⎨
⎪ 2x + 7 y = 18
⎩ Check:
−2x + 3 y = 2 −2(2) + 3(2) = 2
−2x + 3(2) = 2
2x + 7 y = 18 −4 + 6 = 2
−2x + 6 = 2
10 y = 20 −6 −6 2(2) + 7(2) = 18
10 10 −2x = −4 4 + 14 = 18
y=2 −2 −2
x=2 (2, 2)
Sun, Apr 10
33. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪x + 4 y = 5
⎩
Sun, Apr 10
34. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
Sun, Apr 10
35. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
Sun, Apr 10
36. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5
Sun, Apr 10
37. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5
Sun, Apr 10
38. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5
9x = 9
Sun, Apr 10
39. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5
9x = 9
9 9
Sun, Apr 10
40. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5
9x = 9
9 9
x=1
Sun, Apr 10
41. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9
9 9
x=1
Sun, Apr 10
42. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9
x=1
Sun, Apr 10
43. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9 4y = 4
x=1
Sun, Apr 10
44. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9 4y = 4
4 4
x=1
Sun, Apr 10
45. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩(
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9 4y = 4
4 4
x=1
y =1
Sun, Apr 10
46. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5
9x = 9 −1 −1
9 9 4y = 4
4 4
x=1
y =1
Sun, Apr 10
47. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1
9 9 4y = 4
4 4
x=1
y =1
Sun, Apr 10
48. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1 10 + 4 = 14
9 9 4y = 4
4 4
x=1
y =1
Sun, Apr 10
49. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1 10 + 4 = 14
9 9 4y = 4
4 4 1 + 4(1) = 5
x=1
y =1
Sun, Apr 10
50. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1 10 + 4 = 14
9 9 4y = 4
4 4 1 + 4(1) = 5
x=1 1+ 4 = 5
y =1
Sun, Apr 10
51. Example 1
Solve by combining the equations
⎧10x + 4 y = 14
⎪
b. ⎨
⎪ x + 4 y = 5)(−1)
⎩( Check:
10x + 4 y = 14
− x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14
9x = 9 −1 −1 10 + 4 = 14
9 9 4y = 4
4 4 1 + 4(1) = 5
x=1 1+ 4 = 5
y =1
(1,1)
Sun, Apr 10
52. Example 2
Solve by combining the equations
⎧7x + 2 y = 5
⎪
⎨
⎪ 2x + 3 y = 16
⎩
Sun, Apr 10
53. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16
⎩
Sun, Apr 10
54. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
Sun, Apr 10
55. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15
Sun, Apr 10
56. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15
−4x − 6 y = −32
Sun, Apr 10
57. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15
−4x − 6 y = −32
Sun, Apr 10
58. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15
−4x − 6 y = −32
17x = −17
Sun, Apr 10
59. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15
−4x − 6 y = −32
17x = −17
17 17
Sun, Apr 10
60. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15
−4x − 6 y = −32
17x = −17
17 17
x = −1
Sun, Apr 10
61. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32
17x = −17
17 17
x = −1
Sun, Apr 10
62. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17
17 17
x = −1
Sun, Apr 10
63. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17
x = −1
Sun, Apr 10
64. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1
Sun, Apr 10
65. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1 3 3
Sun, Apr 10
66. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1 3 3
y=6
Sun, Apr 10
67. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 −2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1 3 3
y=6
Sun, Apr 10
68. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
17x = −17 +2 +2
17 17 3 y = 18
x = −1 3 3
y=6
Sun, Apr 10
69. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
+2 −7 + 12 = 5
17x = −17 +2
17 17 3 y = 18
x = −1 3 3
y=6
Sun, Apr 10
70. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
+2 −7 + 12 = 5
17x = −17 +2
17 17 3 y = 18 2(−1) + 3(6) = 16
x = −1 3 3
y=6
Sun, Apr 10
71. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
+2 −7 + 12 = 5
17x = −17 +2
17 17 3 y = 18 2(−1) + 3(6) = 16
x = −1 3 3 −2 + 18 = 16
y=6
Sun, Apr 10
72. Example 2
Solve by combining the equations
⎧7x + 2 y = 5)(3)
⎪(
⎨
⎪ 2x + 3 y = 16)(−2)
⎩(
Check:
21x + 6 y = 15 2(−1) + 3 y = 16
−4x − 6 y = −32 7(−1) + 2(6) = 5
−2 + 3 y = 16
+2 −7 + 12 = 5
17x = −17 +2
17 17 3 y = 18 2(−1) + 3(6) = 16
x = −1 3 3 −2 + 18 = 16
y=6
(−1, 6)
Sun, Apr 10
73. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3
⎩
Sun, Apr 10
74. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
Sun, Apr 10
75. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7
Sun, Apr 10
76. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7
−4x − 2 y = 3
Sun, Apr 10
77. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7
−4x − 2 y = 3
Sun, Apr 10
78. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7
−4x − 2 y = 3
x = 10
Sun, Apr 10
79. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3
x = 10
Sun, Apr 10
80. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
x = 10
Sun, Apr 10
81. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10
Sun, Apr 10
82. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10
2 y = −43
Sun, Apr 10
83. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10
2 y = −43
2 2
Sun, Apr 10
84. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10
2 y = −43
2 2
−43
y=
2
Sun, Apr 10
85. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
Check:
5x + 2 y = 7 5(10) + 2 y = 7
−4x − 2 y = 3 50 + 2 y = 7
−50 −50
x = 10
2 y = −43
2 2
−43
y=
2
Sun, Apr 10
86. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 −50
x = 10
2 y = −43
2 2
−43
y=
2
Sun, Apr 10
87. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 50 − 43 = 7
−50
x = 10
2 y = −43
2 2
−43
y=
2
Sun, Apr 10
88. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 50 − 43 = 7
−50
x = 10
2 y = −43 4(10) + 2(− ) = −3
43
2
2 2
−43
y=
2
Sun, Apr 10
89. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 50 − 43 = 7
−50
x = 10
2 y = −43 4(10) + 2(− ) = −3
43
2
2 2 40 − 43 = −3
−43
y=
2
Sun, Apr 10
90. Example 3
Solve by combining the equations
⎧5x + 2 y = 7
⎪
⎨
⎪4x + 2 y = −3)(−1)
⎩(
Check:
5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7
43
−4x − 2 y = 3 50 + 2 y = 7 2
−50 50 − 43 = 7
−50
x = 10
2 y = −43 4(10) + 2(− ) = −3
43
2
2 2 40 − 43 = −3
−43
y=
2
(10, − ) 43
2
Sun, Apr 10