Alg2 lesson 8-7
•
0 likes•220 views
The system of equations is solved as follows: 1) x + 2y = 5 and y = 3x - 1 are substituted into each other and simplified 2) This results in 7x = 7, so x = 1 3) Substituting x = 1 into y = 3x - 1 gives y = 2 4) Therefore, the solution is (1, 2).
Report
Share
Report
Share
Recommended
4th quarter long test review
This document contains a 10 question review for a 4th quarter long test covering circle equations, polynomial division, factoring, and finding zeros of polynomials. The questions cover finding equations of circles given properties like center and radius, performing polynomial division and finding quotients and remainders, factoring polynomials, and solving polynomials for real zeros.
Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4
1. This document discusses calculus formulas for derivatives of common functions including exponential, logarithmic, trigonometric, and implicit functions. It provides the derivative formulas and works through examples of finding derivatives of various functions.
2. Several examples are worked through, applying the formulas to find the derivatives of functions like y = ecos5x, y = (e4x - e5x)4, and implicit functions like x3 + y4 = 0.
3. The document concludes by providing the basic derivative formulas for inverse trigonometric functions and working through an example of finding the derivative of y = arc sin (5 + x2).
0304 ch 3 day 4
The document provides step-by-step work to find the exact zeros of the polynomial function f(x) = 2x^5 - x^4 - 10x^3 + 5x^2 + 12x - 6. It begins by graphing the function to suggest a zero of 0.5 and then uses synthetic division to reduce the polynomial degree. This yields a quadratic function that is then factored to find the zeros of x = ±3 and x = ±2. The final zeros listed are 0.5, ±3, and ±2.
0303 ch 3 day 3
This document discusses finding the rational zeros of polynomials using the Rational Zeros Theorem. It provides examples of finding all rational zeros of polynomials by considering possible values of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. It also discusses using synthetic division and the Quadratic Formula to find the exact zeros of polynomials when not all zeros are rational.
Pt 3&4 turunan fungsi implisit dan cyclometri
This document discusses implicit differentiation and provides examples of taking the derivative of implicit functions. It begins by presenting the general form of an implicit function as f(x,y)=0 and provides some examples. It then shows how to take the derivative dy/dx of several implicit functions by applying the chain rule and implicit differentiation. Formulas for the derivatives of inverse trigonometric functions are also provided, along with examples of finding dy/dx for functions involving inverse trigonometric functions.
Pembahasan Soal Matematika Kelas 10 Semester 1
This document provides the answers to a mathematics exam for 10th grade students. It includes multiple choice questions with explanations and word problems with step-by-step solutions. The topics covered include algebra, logarithms, quadratic equations, and inequalities.
0408 ch 4 day 8
The document provides steps to solve exponential and logarithmic equations:
1. Isolate the exponential expression.
2. Take the log of both sides.
3. Solve and verify all solutions by substitution.
It then works through examples of solving exponential equations, isolating the exponential term, taking logs of both sides, and solving for the variable. Solutions are verified with substitution.
11 X1 T01 08 Simultaneous Equations (2010)
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
Recommended
4th quarter long test review
This document contains a 10 question review for a 4th quarter long test covering circle equations, polynomial division, factoring, and finding zeros of polynomials. The questions cover finding equations of circles given properties like center and radius, performing polynomial division and finding quotients and remainders, factoring polynomials, and solving polynomials for real zeros.
Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4
1. This document discusses calculus formulas for derivatives of common functions including exponential, logarithmic, trigonometric, and implicit functions. It provides the derivative formulas and works through examples of finding derivatives of various functions.
2. Several examples are worked through, applying the formulas to find the derivatives of functions like y = ecos5x, y = (e4x - e5x)4, and implicit functions like x3 + y4 = 0.
3. The document concludes by providing the basic derivative formulas for inverse trigonometric functions and working through an example of finding the derivative of y = arc sin (5 + x2).
0304 ch 3 day 4
The document provides step-by-step work to find the exact zeros of the polynomial function f(x) = 2x^5 - x^4 - 10x^3 + 5x^2 + 12x - 6. It begins by graphing the function to suggest a zero of 0.5 and then uses synthetic division to reduce the polynomial degree. This yields a quadratic function that is then factored to find the zeros of x = ±3 and x = ±2. The final zeros listed are 0.5, ±3, and ±2.
0303 ch 3 day 3
This document discusses finding the rational zeros of polynomials using the Rational Zeros Theorem. It provides examples of finding all rational zeros of polynomials by considering possible values of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. It also discusses using synthetic division and the Quadratic Formula to find the exact zeros of polynomials when not all zeros are rational.
Pt 3&4 turunan fungsi implisit dan cyclometri
This document discusses implicit differentiation and provides examples of taking the derivative of implicit functions. It begins by presenting the general form of an implicit function as f(x,y)=0 and provides some examples. It then shows how to take the derivative dy/dx of several implicit functions by applying the chain rule and implicit differentiation. Formulas for the derivatives of inverse trigonometric functions are also provided, along with examples of finding dy/dx for functions involving inverse trigonometric functions.
Pembahasan Soal Matematika Kelas 10 Semester 1
This document provides the answers to a mathematics exam for 10th grade students. It includes multiple choice questions with explanations and word problems with step-by-step solutions. The topics covered include algebra, logarithms, quadratic equations, and inequalities.
0408 ch 4 day 8
The document provides steps to solve exponential and logarithmic equations:
1. Isolate the exponential expression.
2. Take the log of both sides.
3. Solve and verify all solutions by substitution.
It then works through examples of solving exponential equations, isolating the exponential term, taking logs of both sides, and solving for the variable. Solutions are verified with substitution.
11 X1 T01 08 Simultaneous Equations (2010)
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
Algebra
The document provides 5 algebra word problems and their step-by-step solutions. It begins with a disclaimer that the document was prepared by trainees and is not an official document. It then presents 5 multi-step algebra word problems, showing the work and reasoning for arriving at each solution. The document concludes by providing contact information and a thank you.
Simultaneous equation
This document discusses solving simultaneous linear and quadratic equations. It explains that for a linear equation and a non-linear equation, an unknown can be expressed in terms of the other unknown from the linear equation. This forms a quadratic equation that can then be solved using factorisation or the quadratic formula to obtain the values for both unknowns. As an example, it shows choosing x as the easier unknown from the linear equation x+2y=4 to get x=4-2y, then substituting this into the quadratic equation x^2+xy+y^2=7. This results in a quadratic equation that can be factorised to solve for y and back substitute to find x.
Simultaneous equations (2)
The document discusses solving systems of 3 linear equations with 3 unknowns. It provides examples of using the elimination method, which involves rewriting the system as two smaller systems, eliminating the same variable from each, solving the resulting system of 2 equations for the remaining 2 variables, then substituting back into one of the original equations to find the third variable. The solution is written as an ordered triple (x, y, z). It demonstrates this process on examples and encourages practicing this method.
The Mr. K question
The document describes a plan to distract a teacher, Mr. K, in order to steal his coffee. It involves throwing his block of wood in the hallway so he would discover a smart board with tricky math questions. While Mr. K was focused on fixing errors in the answers, the students were able to steal his coffee. The smart board then provides the correct solutions to the math questions to further distract Mr. K.
Quadratic eq and discriminant
The document discusses the quadratic formula and how to use it to solve quadratic equations. It introduces the quadratic formula as x = (-b ±√(b2 - 4ac))/2a and explains how to use it to solve equations by plugging in the a, b, and c coefficients. It also defines the discriminant b2 - 4ac and explains what types of roots (real, complex, rational, irrational) an equation will have depending on whether the discriminant is positive, negative, or zero. Several examples are worked through to demonstrate applying the quadratic formula.
Factorización aplicando Ruffini o Método de Evaluación
1. The document discusses factorizing polynomials using Ruffini's method.
2. It provides two examples of factorizing polynomials into their prime factors: (1) x3 + 5x2 - 2x - 24 = (x-2)(x+3)(x+4) and (2) 2x3 - 3x2 - 11x + 6 = (2x - 1)(x + 2)(x - 3).
3. For each polynomial, it lists the possible factors of the constant term and coefficients, then uses those factors to find the roots that allow factoring the polynomial.
Ecuaciones
The document contains examples of simple first-degree equations with:
- No parentheses or denominators
- Terms grouped together
- Parentheses
The equations are solved for the variable x, with the solutions provided.
Factors of po lynomials + solving equations
This document discusses factorizing polynomials of degree 3 or higher using the factor theorem or "The Big L" method. It provides examples of factorizing polynomials and using the factors to find the roots or solutions of polynomial equations. The examples show setting a polynomial equal to 0, finding a factor using the factor theorem, fully factorizing the polynomial, and then setting each factor equal to 0 to obtain the roots. The document emphasizes that the factor theorem can be used to determine if an expression is a factor if the remainder is 0 upon dividing the polynomial by the expression.
MODULE 4- Quadratic Expression and Equations
(1) The document is a math worksheet containing 20 quadratic equations to solve.
(2) It provides the steps to solve each equation, factorizing the expressions and setting each factor equal to zero to find the roots.
(3) The answers section lists the factored forms and solutions for each of the 20 equations.
Sketching the graph of a polynomial function
This document discusses how to find the x-intercepts and y-intercepts of a polynomial function. To find the x-intercepts, one must factor the polynomial completely and set each factor equal to zero to solve for x. To find the y-intercept, set x equal to zero and simplify the polynomial. For the example polynomial function x^4 - 5x^2 + 4, the x-intercepts are (1,0), (-1,0), (2,0), (-2,0) and the y-intercept is (0,4).
Quadratic And Roots
This document discusses quadratic equations and their properties. It defines quadratic equations as equations of the form y=ax^2 +bx + c, where the highest power is 2. It explains that quadratic equations can be solved using the quadratic formula, x = -b ± √(b^2 - 4ac) / 2a. The number of solutions depends on the discriminant, b^2 - 4ac. If it is greater than 0, there are two solutions, if equal to 0 there is one solution, and if less than 0 there are no solutions. Examples are provided to demonstrate solving quadratic equations.
Simultaneous equations
The document discusses solving simultaneous equations. It provides examples of simultaneous equations involving two variables (x and y) and two equations, including one linear and one non-linear equation. Methods for solving the simultaneous equations include expressing one variable in terms of the other, substituting one equation into the other, and solving for the variables. Solutions may have multiple answer pairs for x and y.
Quadratic Equations (Quadratic Formula) Using PowerPoint
This document summarizes the steps to solve a quadratic equation using the quadratic formula. It works through solving the specific equation 5y^2 - 8y + 3 = 0 as an example. The key steps are: 1) Identifying the coefficients a, b, and c; 2) Plugging these into the quadratic formula; 3) Simplifying the terms; 4) Isolating the variable to find the solutions of 1 and 0.6. These solutions are then checked by substituting them back into the original equation.
Alg2 lesson 3-5
1) The document describes different geometric figures based on systems of equations in one, two, and three dimensions.
2) It provides examples of solving systems of equations in two and three variables, eliminating variables to solve the systems.
3) The solutions provided are the point (-2, 6, -3) for a system in three variables, and that some systems have infinite solutions or no solution.
Factorising quadratic expressions 1
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
Funciones1
1. The document describes methods for completing squares to graph functions and find their domains and ranges.
2. Examples are provided of using the method to graph several functions and find their corresponding domains and ranges.
3. For each function, the method involves completing the square of the variables, finding the x- and y-intercepts, and stating the domain and range.
Quadratic Equation and discriminant
This document introduces the quadratic formula as a method for solving quadratic equations. It shows the steps for deriving the formula from completing the square and provides examples of its use. The discriminant is defined as b^2 - 4ac from the quadratic formula. The sign of the discriminant determines the number and type of roots: positive discriminant yields two real roots, zero discriminant yields one real root, and negative discriminant yields two complex roots. Examples are provided to illustrate each case.
Topic 14 algebra
The document discusses rules for indices and factorizing algebraic expressions. It provides examples of:
- Multiplying and dividing terms with the same base using index rules like ab × ac = ab+c and ab ÷ ac = ab-c.
- Expanding single and double brackets by distributing terms.
- Finding common factors to group like terms.
- Factorizing quadratics and using differences of squares.
- Solving equations set equal to zero by factorizing.
Factoring polynomials
Factoring polynomials involves finding common factors that can be divided out of terms, similar to factoring numbers but with variables; this is done by looking for a single variable or number that is a common factor of all terms that can be pulled out in front of parentheses. The document provides examples of different types of factoring polynomials including using the greatest common factor, difference of squares, grouping, and perfect squares and cubes.
The discriminant
The document discusses the discriminant and how it is used to determine the number of solutions to a quadratic equation. The discriminant, b^2 - 4ac, is calculated and compared to 0 to find the number of solutions: if greater than 0 there are 2 solutions, if equal to 0 there is 1 solution, if less than 0 there are no real solutions. Several examples are shown of calculating the discriminant and determining the nature of the roots. The document also discusses how to find the value of a variable if the roots are equal.
More Related Content
What's hot
Algebra
The document provides 5 algebra word problems and their step-by-step solutions. It begins with a disclaimer that the document was prepared by trainees and is not an official document. It then presents 5 multi-step algebra word problems, showing the work and reasoning for arriving at each solution. The document concludes by providing contact information and a thank you.
Simultaneous equation
This document discusses solving simultaneous linear and quadratic equations. It explains that for a linear equation and a non-linear equation, an unknown can be expressed in terms of the other unknown from the linear equation. This forms a quadratic equation that can then be solved using factorisation or the quadratic formula to obtain the values for both unknowns. As an example, it shows choosing x as the easier unknown from the linear equation x+2y=4 to get x=4-2y, then substituting this into the quadratic equation x^2+xy+y^2=7. This results in a quadratic equation that can be factorised to solve for y and back substitute to find x.
Simultaneous equations (2)
The document discusses solving systems of 3 linear equations with 3 unknowns. It provides examples of using the elimination method, which involves rewriting the system as two smaller systems, eliminating the same variable from each, solving the resulting system of 2 equations for the remaining 2 variables, then substituting back into one of the original equations to find the third variable. The solution is written as an ordered triple (x, y, z). It demonstrates this process on examples and encourages practicing this method.
The Mr. K question
The document describes a plan to distract a teacher, Mr. K, in order to steal his coffee. It involves throwing his block of wood in the hallway so he would discover a smart board with tricky math questions. While Mr. K was focused on fixing errors in the answers, the students were able to steal his coffee. The smart board then provides the correct solutions to the math questions to further distract Mr. K.
Quadratic eq and discriminant
The document discusses the quadratic formula and how to use it to solve quadratic equations. It introduces the quadratic formula as x = (-b ±√(b2 - 4ac))/2a and explains how to use it to solve equations by plugging in the a, b, and c coefficients. It also defines the discriminant b2 - 4ac and explains what types of roots (real, complex, rational, irrational) an equation will have depending on whether the discriminant is positive, negative, or zero. Several examples are worked through to demonstrate applying the quadratic formula.
Factorización aplicando Ruffini o Método de Evaluación
1. The document discusses factorizing polynomials using Ruffini's method.
2. It provides two examples of factorizing polynomials into their prime factors: (1) x3 + 5x2 - 2x - 24 = (x-2)(x+3)(x+4) and (2) 2x3 - 3x2 - 11x + 6 = (2x - 1)(x + 2)(x - 3).
3. For each polynomial, it lists the possible factors of the constant term and coefficients, then uses those factors to find the roots that allow factoring the polynomial.
Ecuaciones
The document contains examples of simple first-degree equations with:
- No parentheses or denominators
- Terms grouped together
- Parentheses
The equations are solved for the variable x, with the solutions provided.
Factors of po lynomials + solving equations
This document discusses factorizing polynomials of degree 3 or higher using the factor theorem or "The Big L" method. It provides examples of factorizing polynomials and using the factors to find the roots or solutions of polynomial equations. The examples show setting a polynomial equal to 0, finding a factor using the factor theorem, fully factorizing the polynomial, and then setting each factor equal to 0 to obtain the roots. The document emphasizes that the factor theorem can be used to determine if an expression is a factor if the remainder is 0 upon dividing the polynomial by the expression.
MODULE 4- Quadratic Expression and Equations
(1) The document is a math worksheet containing 20 quadratic equations to solve.
(2) It provides the steps to solve each equation, factorizing the expressions and setting each factor equal to zero to find the roots.
(3) The answers section lists the factored forms and solutions for each of the 20 equations.
Sketching the graph of a polynomial function
This document discusses how to find the x-intercepts and y-intercepts of a polynomial function. To find the x-intercepts, one must factor the polynomial completely and set each factor equal to zero to solve for x. To find the y-intercept, set x equal to zero and simplify the polynomial. For the example polynomial function x^4 - 5x^2 + 4, the x-intercepts are (1,0), (-1,0), (2,0), (-2,0) and the y-intercept is (0,4).
Quadratic And Roots
This document discusses quadratic equations and their properties. It defines quadratic equations as equations of the form y=ax^2 +bx + c, where the highest power is 2. It explains that quadratic equations can be solved using the quadratic formula, x = -b ± √(b^2 - 4ac) / 2a. The number of solutions depends on the discriminant, b^2 - 4ac. If it is greater than 0, there are two solutions, if equal to 0 there is one solution, and if less than 0 there are no solutions. Examples are provided to demonstrate solving quadratic equations.
Simultaneous equations
The document discusses solving simultaneous equations. It provides examples of simultaneous equations involving two variables (x and y) and two equations, including one linear and one non-linear equation. Methods for solving the simultaneous equations include expressing one variable in terms of the other, substituting one equation into the other, and solving for the variables. Solutions may have multiple answer pairs for x and y.
Quadratic Equations (Quadratic Formula) Using PowerPoint
This document summarizes the steps to solve a quadratic equation using the quadratic formula. It works through solving the specific equation 5y^2 - 8y + 3 = 0 as an example. The key steps are: 1) Identifying the coefficients a, b, and c; 2) Plugging these into the quadratic formula; 3) Simplifying the terms; 4) Isolating the variable to find the solutions of 1 and 0.6. These solutions are then checked by substituting them back into the original equation.
Alg2 lesson 3-5
1) The document describes different geometric figures based on systems of equations in one, two, and three dimensions.
2) It provides examples of solving systems of equations in two and three variables, eliminating variables to solve the systems.
3) The solutions provided are the point (-2, 6, -3) for a system in three variables, and that some systems have infinite solutions or no solution.
Factorising quadratic expressions 1
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
Funciones1
1. The document describes methods for completing squares to graph functions and find their domains and ranges.
2. Examples are provided of using the method to graph several functions and find their corresponding domains and ranges.
3. For each function, the method involves completing the square of the variables, finding the x- and y-intercepts, and stating the domain and range.
Quadratic Equation and discriminant
This document introduces the quadratic formula as a method for solving quadratic equations. It shows the steps for deriving the formula from completing the square and provides examples of its use. The discriminant is defined as b^2 - 4ac from the quadratic formula. The sign of the discriminant determines the number and type of roots: positive discriminant yields two real roots, zero discriminant yields one real root, and negative discriminant yields two complex roots. Examples are provided to illustrate each case.
Topic 14 algebra
The document discusses rules for indices and factorizing algebraic expressions. It provides examples of:
- Multiplying and dividing terms with the same base using index rules like ab × ac = ab+c and ab ÷ ac = ab-c.
- Expanding single and double brackets by distributing terms.
- Finding common factors to group like terms.
- Factorizing quadratics and using differences of squares.
- Solving equations set equal to zero by factorizing.
Factoring polynomials
Factoring polynomials involves finding common factors that can be divided out of terms, similar to factoring numbers but with variables; this is done by looking for a single variable or number that is a common factor of all terms that can be pulled out in front of parentheses. The document provides examples of different types of factoring polynomials including using the greatest common factor, difference of squares, grouping, and perfect squares and cubes.
The discriminant
The document discusses the discriminant and how it is used to determine the number of solutions to a quadratic equation. The discriminant, b^2 - 4ac, is calculated and compared to 0 to find the number of solutions: if greater than 0 there are 2 solutions, if equal to 0 there is 1 solution, if less than 0 there are no real solutions. Several examples are shown of calculating the discriminant and determining the nature of the roots. The document also discusses how to find the value of a variable if the roots are equal.
What's hot (20)
Factorización aplicando Ruffini o Método de Evaluación
Factorización aplicando Ruffini o Método de Evaluación
Quadratic Equations (Quadratic Formula) Using PowerPoint
Quadratic Equations (Quadratic Formula) Using PowerPoint
Viewers also liked
OpenAIRE workshops Overview – Najla Rettberg
OpenAIRE Interoperability Workshop (8 Feb. 2013).
OpenAIRE workshops Overview – Najla Rettberg, OpenAIREplus.
Contents page
This document outlines the stages of development for a magazine contents page layout. It describes changes made to colors, images, and elements at each stage. The final stage includes additional band and story listings to suggest more content beyond the main feature article and uses a consistent color scheme to tie the contents page visually to the rest of the magazine. The creator is pleased with the outcome, finding it was worth the time and effort to finalize.
Soal un kimia sma 2014 8
Soal kimia tentang senyawa organik dan anorganik, sifat koloid, reaksi asam basa, dan polimer. Terdiri dari 23 pertanyaan pilihan ganda.
Viewers also liked (8)
Similar to Alg2 lesson 8-7
Simultaneous equations
The document discusses methods for solving simultaneous linear equations, including elimination and substitution.
It provides examples of using elimination by adding or subtracting equations to remove a variable, and substitution by making one variable the subject of an equation and substituting it into the other equation. Fractions are converted to simple linear equations by finding a common denominator. The document also covers solving simultaneous equations when one equation is quadratic using substitution after making one variable the subject of the linear equation.
7.4
This document provides examples of solving systems of nonlinear equations in two variables. It begins with definitions, including that a nonlinear system contains at least one equation that is not of the form Ax + By = C. Methods for solving nonlinear systems include substitution and addition. Examples walk through both methods step-by-step for various systems. Key steps are rewriting equations in terms of variables, substituting values, solving resulting equations, back-substituting, and checking solutions satisfy both original equations.
Lecture 11 systems of nonlinear equations
The document discusses solving systems of nonlinear equations in two variables. It provides examples of nonlinear systems that contain equations that are not in the form Ax + By = C, such as x^2 = 2y + 10. Methods for solving nonlinear systems include substitution and addition. The substitution method involves solving one equation for one variable and substituting into the other equation. The addition method involves rewriting the equations and adding them to eliminate variables. Examples demonstrate both methods and finding the solution set that satisfies both equations.
Straight-Line-Graphs-Final -2.pptx
This document contains 5 math problems involving factorizing expressions, solving equations, evaluating expressions for given values, expanding expressions, and finding the highest common factor. It also provides context on working with straight line graphs, including finding the gradient and y-intercept of a line from its equation, finding the gradient between two points, finding the midpoint and a point that divides a line segment in a given ratio, and finding the x- and y-intercepts of a line.
Topic 8 (Writing Equations Of A Straight Lines)
The document provides information on properties of straight lines and methods to write equations of straight lines given different conditions. It also discusses solving systems of linear equations using different methods like graphing, substitution, elimination and Cramer's rule. Key points covered include writing equations in slope-intercept and standard form, finding slopes of parallel and perpendicular lines, and properties of consistent, inconsistent and dependent systems of linear equations.
Sect1 1
This section introduces differential equations and their use in mathematical modeling. It provides examples of verifying solutions to differential equations by direct substitution. Typical problems show finding an integrating constant to satisfy an initial condition. Differential equations are derived from descriptions of real-world phenomena involving rates of change. The section establishes foundational knowledge of differential equations and their solution methods.
Lesson 14 a - parametric equations
Parametric equations define a curve where x and y are defined in terms of a third variable called a parameter. The graph of parametric equations consists of all points (x,y) obtained by allowing the parameter to vary over its domain. Eliminating the parameter between the two equations yields a non-parametric equation of the curve. Examples are provided of eliminating parameters between various parametric equations to obtain the curve and sketching the resulting graphs. Exercises are given to further practice eliminating parameters and sketching curves.
Linear systems with 3 unknows
This document provides examples for solving systems of linear equations in three variables. It begins with an example using elimination to solve the system 5x - 2y - 3z = -7, etc. step-by-step, reducing it to a 2x2 system and solving for x, y, and z. The next example uses substitution to solve a word problem about ticket sales. It shows setting up and solving a 3x3 system. The document concludes with an example of a system having an infinite number of solutions.
Module 2 polynomial functions
This module discusses methods for finding the zeros of polynomial functions of degree greater than 2, including: factor theorem, factoring, synthetic division, and depressed equations. It introduces the number of roots theorem, which states that a polynomial of degree n has n roots. It also discusses determining the rational zeros of a polynomial using the rational roots theorem and factor theorem. Examples are provided to illustrate these concepts and methods.
Ml lesson 4 2
The document discusses graphing linear equations using x-y charts. It provides examples of solving single-variable and two-variable linear equations for x and y. Specific examples shown include finding the solution sets for various equations and determining if given points satisfy particular linear equations. Charts are completed and equations are graphed.
Form 4 Add Maths Note
This document provides notes on functions and quadratic equations from Additional Mathematics Form 4. It includes:
1) Definitions of functions, including function notation f(x) and the relationship between objects and images.
2) Methods for solving quadratic equations, including factorisation, completing the square, and the quadratic formula.
3) Properties of quadratic functions like finding the maximum/minimum value and sketching the graph.
4) Solving simultaneous equations involving one linear and one non-linear equation through substitution.
5) Conversions between index and logarithmic forms and basic logarithm laws.
Form 4-add-maths-note
This document provides notes on key concepts in additional mathematics including:
1) Functions such as f(x) = x + 3 and finding the object and image of a function.
2) Solving quadratic equations using factorisation and the quadratic formula. Types of roots are discussed.
3) Sketching quadratic functions by finding the y-intercept, maximum/minimum values, and a third point. Quadratic inequalities are also covered.
4) Methods for solving simultaneous equations including substitution when one equation is nonlinear.
5) Properties of exponents and logarithms, and how to solve exponential and logarithmic equations.
Advanced Engineering Mathematics Solutions Manual.pdf
This document contains 27 multi-part exercises involving differential equations. The exercises cover topics such as determining whether differential equations are linear or nonlinear, solving differential equations, and classifying differential equations by order.
Mathematics 8 Systems of Linear Inequalities
This learner's module discusses and help the students about the topic Systems of Linear Inequalities. It includes definition, examples, applications of Systems of Linear Inequalities.
Chapter 6 taylor and maclaurin series
This document provides examples of finding Taylor and Maclaurin series expansions for various functions. It gives the step-by-step workings for finding the first few terms of series expansions centered at different points for functions like ln(x), 1/x, sin(x), x^4 + x^2, (x-1)e^x, and others. It also discusses using these expansions to approximate integrals and find sums of infinite series.
CIRCLES.pptx
Circles are geometric shapes defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius. Circle equations can be written in standard form (x-h)2 + (y-k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. Problems involve finding the equation of a circle given its center and radius, graphing circles, rewriting equations in standard form, determining if an equation represents a circle, point or null set, finding equations of circles tangent to lines or passing through points, and finding equations of circles inscribed in or
CalculusStudyGuide
The document provides examples and explanations of concepts related to calculus including continuity, differentiability, limits, derivatives, and the mean value theorem. Some key points:
- It gives examples of determining if functions are continuous and differentiable at various points, including functions with absolute value.
- The mean value theorem is explained and examples are worked through showing a function satisfies the mean value theorem and finding the value of c.
- Numerous examples demonstrate calculating derivatives using rules like product, quotient, chain and implicit differentiation. Examples include derivatives of trigonometric, exponential and logarithmic functions.
- Implicit differentiation is used to find the equation of a tangent line to a curve at a given point.
Maths05
The document provides solutions to questions from an IIT-JEE mathematics exam. It includes 8 questions worth 2 marks each, 8 questions worth 4 marks each, and 2 questions worth 6 marks each. The solutions solve problems related to probability, trigonometry, geometry, calculus, and loci. The summary focuses on the high-level structure and content of the document.
Assignment For Matlab Report Subject Calculus 2
This document provides the requirements and assignments for a Calculus 2 Matlab report. It includes topics such as: finding partial derivatives of various functions, studying extrema of functions, evaluating double and triple integrals, and calculating mass and centers of mass of solids. Students are divided into groups and will be randomly assigned a topic involving solving concrete problems numerically using Matlab.
Graphing lines
The two lines y = -3x and 6x + 2y = 4 are parallel because they have the same slope of -3 but different y-intercepts of 0 and 2 respectively. This is shown by rewriting the equations in slope-intercept form, making a table of values for each equation with the same x-values, and noticing that the difference between the y-values is always 2. Plotting the points on a graph also shows the lines with the same slope running parallel without intersecting.
Similar to Alg2 lesson 8-7 (20)
Advanced Engineering Mathematics Solutions Manual.pdf
Advanced Engineering Mathematics Solutions Manual.pdf
More from Carol Defreese
1150 day 11
Ratios, proportions, and percents can be used to compare quantities and solve problems. A ratio compares two quantities, like the number of squares to circles. Equal ratios form a proportion, which can be used to solve for unknown values. Percents represent a number out of 100. Common percent calculations include finding a percent of a number and converting between fractions, decimals, and percents. The percent proportion states that the percent equals the part divided by the whole quantity.
1150 day 10
This document covers topics related to repeating decimals, irrational numbers, and square roots. It includes examples of writing repeating decimals as fractions, ordering decimals, finding square roots, and using the Pythagorean theorem. Various math problems are presented along with step-by-step solutions to illustrate these concepts.
1150 day 9
1) Decimals represent numbers using place value with decimal points. To write decimals in expanded form, they are broken into place value terms using positive and negative exponents.
2) Fractions can be converted to decimals using division or by writing the fraction as a ratio of two numbers and setting up a proportion to solve for the decimal.
3) Scientific notation is used to write very large or small numbers in a standard form, such as 3.603 x 107. Operations can be done on decimals by lining up the decimal points or moving them with multiplication/division.
1150 day 8
This document provides an overview of rational numbers including:
- Integers and fractions written in the form a/b where a and b are integers and b ≠ 0
- Equivalent fractions represented by the same number
- Ordering and comparing rational numbers
- Converting between improper fractions and mixed numbers
- Basic operations of addition, subtraction, multiplication, and division of rational numbers
- Word problems involving rational numbers
1150 day 7
This document discusses prime and composite numbers, greatest common divisors (GCD), and least common multiples (LCM). It provides examples of finding the GCD and LCM of various numbers using different methods like the intersection of sets method, prime factorization method, and Euclidean algorithm. Key definitions include: a prime number has exactly two distinct positive divisors, a composite number has factors other than itself and 1, the GCD is the largest integer that divides numbers, and the LCM is the smallest number that is a multiple of the given numbers.
1150 day 6
Integers include positive and negative whole numbers. Absolute value is the distance from zero. Addition and subtraction on a number line involve moving left or right. Multiplication follows patterns based on sign. Division is the inverse of multiplication regarding sign. A number is divisible by another if it can be written as a product of an integer multiple. Divisibility tests identify patterns in a number's digits.
1150 day 5
Algebraic thinking involves recognizing patterns, modeling situations with symbols, and analyzing change. It relies on understanding variables to represent unknown quantities. The document traces the evolution of algebraic thinking from simple equations to more complex concepts like functions, composite functions, and properties of equations. It provides examples of how algebraic reasoning and symbols can be used to represent and solve real-world problems.
1150 day 4
The document discusses various aspects of whole number operations including:
1) Place value systems and how numbers are represented in bases other than 10 such as base 5 and base 12.
2) Algorithms for addition, subtraction, multiplication, and division using various models and representations.
3) Properties of operations like the commutative, associative, identity, and zero properties of multiplication.
1150 day 3
The document discusses different number systems used throughout history including Hindu-Arabic, Egyptian, Babylonian, Mayan, and Roman systems. It also covers basic concepts of whole numbers such as addition, subtraction, and their properties. Different models are presented to demonstrate whole number operations including set, number line, take-away, comparison, and missing addend models.
1150 day 2
The document discusses sets and set operations including defining sets, elements, cardinal and ordinal numbers, equal and equivalent sets, subsets, Venn diagrams, and set operations like union, intersection, and complement. Examples are provided to illustrate concepts like finite and infinite sets, subsets, Venn diagrams representing multiple sets and operations, and using Venn diagrams to solve problems involving sets and their relationships.
1150 day 1
The number of terms in each sequence can be determined by finding a one-to-one correspondence with the counting numbers.
Alg2 lesson 13-5
This document discusses solving an ambiguous triangle where it is not a right triangle and the measures of angles and sides are not fully known. It determines that trigonometric functions and the Law of Sines cannot be used. It identifies that the Law of Cosines is the appropriate theorem to use to begin solving since it requires only knowing the measures of three sides.
Alg2 lesson 13-4
The document discusses solving for unknown sides and angles of triangles using trigonometric functions like the law of sines. Several multi-step example problems are worked through that involve determining if a triangle is possible based on given information, finding missing side lengths, and calculating angles. Diagrams are included to illustrate each step of the example problems.
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
5) Finding the value of sin(135°)
6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
5) Finding the value of sin(135°)
6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-2
The document describes how to draw angles in standard position on a unit circle and convert between degrees and radians. It provides examples of drawing angles such as 210°, -45°, and 540° in standard position and rewriting angles such as 30°, 45°, and an unspecified angle in radians and degrees. It also gives examples of finding coterminal angles for 210° with one positive and one negative measure.
Alg2 lesson 13-1
This document discusses trigonometric functions and their ratios, including sine, cosine, tangent, cosecant, secant, and cotangent. It provides examples of using trigonometric functions to solve values in right triangles, including finding missing side lengths and angle measures. Special right triangles with ratios of 1/2, 1/√3, and 1/√2 are also covered.
Alg2 lesson 11-2 and 11-4
The document provides information about arithmetic and geometric series. It defines arithmetic and geometric series, provides examples of finding sums of arithmetic series using formulas, and defines the key terms (first term, common ratio, number of terms, last term) used in the formula to calculate the sum of a geometric series.
Alg2 lesson 11-3
The document discusses geometric sequences, which are patterns of numbers where each term is found by multiplying the previous term by a constant called the common ratio. It provides examples of geometric sequences with different common ratios and shows how to write equations to find the nth term in a sequence. It also explains how to find specific terms in a geometric sequence and calculates the geometric means between two nonconsecutive terms.
Alg2 lesson 11-1
The document discusses arithmetic sequences, which are patterns of numbers where each term is found by adding a constant value to the previous term. It provides examples of arithmetic sequences with different common differences and shows how to write equations to determine the nth term. It also explains how to find terms within an arithmetic sequence and determine arithmetic means between two non-consecutive terms.
More from Carol Defreese (20)
Recently uploaded
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
How to Download & Install Module From the Odoo App Store in Odoo 17
Custom modules offer the flexibility to extend Odoo's capabilities, address unique requirements, and optimize workflows to align seamlessly with your organization's processes. By leveraging custom modules, businesses can unlock greater efficiency, productivity, and innovation, empowering them to stay competitive in today's dynamic market landscape. In this tutorial, we'll guide you step by step on how to easily download and install modules from the Odoo App Store.
How to Fix [Errno 98] address already in use
This slide will represent the cause of the error “[Errno 98] address already in use” and the troubleshooting steps to resolve this error in Odoo.
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.ppt
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
KHUSWANT SINGH.pptx ALL YOU NEED TO KNOW ABOUT KHUSHWANT SINGH
INDIA`S OWN LITERARY GENIUS MR.KHUSHWANT SINGH WAS TRULY A VERY BRAVE SOUL AND WAS AWARDED WITH THE MAGIC OF WORDS BY GOD.
Wound healing PPT
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
CapTechTalks Webinar Slides June 2024 Donovan Wright.pptx
Slides from a Capitol Technology University webinar held June 20, 2024. The webinar featured Dr. Donovan Wright, presenting on the Department of Defense Digital Transformation.
How to Manage Reception Report in Odoo 17
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
Temple of Asclepius in Thrace. Excavation results
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Contiguity Of Various Message Forms - Rupam Chandra.pptx
Contiguity Of Various Message Forms - Rupam Chandra.pptx
HYPERTENSION - SLIDE SHARE PRESENTATION.
IT WILL BE HELPFULL FOR THE NUSING STUDENTS
IT FOCUSED ON MEDICAL MANAGEMENT AND NURSING MANAGEMENT.
HIGHLIGHTS ON HEALTH EDUCATION.
Recently uploaded (20)
SWOT analysis in the project Keeping the Memory @live.pptx
SWOT analysis in the project Keeping the Memory @live.pptx
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...
How to Download & Install Module From the Odoo App Store in Odoo 17
How to Download & Install Module From the Odoo App Store in Odoo 17
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.ppt
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.ppt
KHUSWANT SINGH.pptx ALL YOU NEED TO KNOW ABOUT KHUSHWANT SINGH
KHUSWANT SINGH.pptx ALL YOU NEED TO KNOW ABOUT KHUSHWANT SINGH
REASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdf
REASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdf
spot a liar (Haiqa 146).pptx Technical writhing and presentation skills
spot a liar (Haiqa 146).pptx Technical writhing and presentation skills
CapTechTalks Webinar Slides June 2024 Donovan Wright.pptx
CapTechTalks Webinar Slides June 2024 Donovan Wright.pptx
Juneteenth Freedom Day 2024 David Douglas School District
Juneteenth Freedom Day 2024 David Douglas School District
Contiguity Of Various Message Forms - Rupam Chandra.pptx
Contiguity Of Various Message Forms - Rupam Chandra.pptx
Alg2 lesson 8-7
- 1. Solve the system of equations: x + 2y = 5 y = 3x – 1 x + 2(3x – 1) = 5 y = 3x – 1 x + 6x – 2 = 5 y = 3(1) – 1 7x = 7 y=2 x=1 (1, 2)
- 2. Solve the system of equations: x + 2y = 5 y = 3x - 1 (1,2)
- 3. circle and parabola 0 intersections
- 4. circle and parabola 1 intersection
- 5. circle and parabola 2 intersections
- 6. circle and parabola 3 intersections
- 7. circle and parabola 4 intersections
- 8. 2 circles 0 intersections
- 9. 2 circles 1 intersection
- 10. 2 circles 2 intersections
- 11. circle and ellipse 0 intersections
- 12. circle and ellipse 1 intersection
- 13. circle and ellipse 2 intersections
- 14. circle and ellipse 3 intersections
- 15. circle and ellipse 4 intersections
- 16. Solve the system of equations. hyperbola line 4(4 – 8y + 4y2) – 16y2 = 25 16 – 32y + 16y2 – 16y2 = 25
- 18. Solve the system of equations. circle y2 = 10 – x2 ellipse if x = 3: 3x2 + 10 – x2 = 28 y2 =10 – (3)2 y2 = 1 2x2 = 18 y= 1 x2 = 9 if x = -3: x = 3 y2 =10 – (-3)2 y2 = 1 y= 1 Answer: (3, 1), (3, –1) , (–3, 1), and (–3, –1)
- 19. Answer: (3, 1), (3, –1) , (–3, 1), and (–3, –1)