Lesson 7.1
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The document discusses linear systems of equations. A linear system is a set of two or more equations with the same variables where the graph of each equation is a line. Linear systems can have one solution, infinite solutions, or no solution depending on whether the lines intersect, are the same, or are parallel.
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A system of linear equations is a set of two or more linear equations with the same variables. There are three types of systems: independent systems have different slopes and intercepts with a single solution point; dependent systems have equations representing the same line with an infinite number of solutions; inconsistent systems have parallel lines with no solution since the lines never intersect. Systems can be solved graphically by plotting the lines and finding their intersection points, or algebraically by elimination of variables.
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7.5 notes[1]
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7.5 notes[1]
The document discusses different methods for solving linear systems, including graphing, substitution, addition, subtraction, and multiplication. It explains that a linear system can have one solution (consistent independent), no solution (inconsistent), or infinitely many solutions (consistent dependent), depending on whether the lines are parallel, intersect at a single point, or are the same line. Examples are provided to illustrate determining the number of solutions based on the slopes and y-intercepts when the equations are in slope-intercept form.
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* Identify inconsistent systems of equations containing two variables.
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The type of solution can be determined without graphing by writing the equations in slope-intercept form and comparing slopes and y-intercepts. Different slopes means one solution, same slope but different intercepts means no solution, and same slope and intercept means infinite solutions.
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Here are the solutions to the systems of equations without graphing:
1. One solution (different slopes)
2. No solution (same slope, different intercepts)
3. One solution (different slopes)
4. No solution (same slope, different intercepts)
5. One solution (different slopes)
6. Infinitely many solutions (same slope and intercept)
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Section 1.5
Parametric equations allow functions to be defined using a third variable t, called a parameter, to define both x and y values. For example, a function can be defined parametrically as x=f(t) and y=g(t). Inverse functions relate each element of an ordered pair (x,y) to another ordered pair by reflecting points across the line y=x. This is known as the inverse reflection principle. Inverse functions satisfy the horizontal line test, meaning each x value corresponds to only one y value.
2.5 2.6 lesson
The document discusses applying the distributive property to algebraic expressions by breaking them down into terms, coefficients, and constants and then combining like terms. It provides examples of distributing multiplication over addition in expressions like 4(y+3), n(n-9), and -2(x+7). It also covers simplifying expressions involving subtraction, multiplication of terms with variables and numbers, and division of real numbers. The homework assigned is to complete odd problems from pages 99-100 and all problems from pages 106-107.
2.1 lesson
This document discusses different types of real numbers including rational numbers, irrational numbers, integers, and whole numbers. It defines rational numbers and discusses their properties such as opposites and absolute value. It also explains what conditional statements are, noting they have a hypothesis and conclusion in an "if-then" format and can be proven true or false with a counterexample. Examples of conditional statements involving rational numbers and absolute value are provided. Students are assigned practice problems from lessons 2.2, 2.3, and 2.4.
Section 1.3
This document discusses graphing the 12 basic functions in mathematics. It provides an example of graphing the 12 basic functions and mentions that graphing these functions is an example. The document repeats the phrase "Graph the 12 Basic Functions" three times.
Alg 2 sections 2.1
This document discusses relations and functions. It defines a relation as a mapping or pairing of input and output values, with the input set called the domain and output set called the range. Relations can be represented through tables, mapping diagrams, ordered pairs, and graphs. The document defines a function as a special type of relation where each input has exactly one output; if any input has more than one output it is not a function. Examples of functions and non-functions are provided. Homework problems from the textbook are listed at the end.
Section 1.1
The document discusses mathematical modeling and functions. It defines a mathematical model as a structure that approximates real-world phenomena to study or predict behavior. It then discusses numerical, algebraic and graphical models. It also discusses functions, including defining them, representing them with equations, graphs and mappings, and determining their domains and ranges. It provides examples of finding domains and ends with homework problems assigned.
Section 1.1
The document discusses mathematical modeling and functions. It defines mathematical modeling as using mathematical structures to approximate real-world phenomena in order to study or predict behavior. It then discusses different types of mathematical models like numerical, algebraic and graphical models. The rest of the document focuses on defining and representing functions, including discussing domain and range, and ways to find the domain of a function either algebraically or graphically. It concludes by listing specific homework problems.
Alg2 section 1.7
This document discusses how to solve absolute value equations and inequalities in 3 sections: the first section explains absolute value, the second section provides instructions for solving absolute value equations, and the third section gives directions for solving absolute value inequalities and includes an application example.
Pa section 1.6
The document defines and provides examples of calculating the median, mode, and range of data sets. It explains that the median is the middle value of a data set in numerical order, or the mean of the two middle values for an even number of values. The mode is the value that occurs most frequently. The range is the difference between the greatest and least values. Examples are provided to demonstrate calculating each metric for temperature and score data sets.
Pa section 1.6
The document defines and provides examples of calculating the median, mode, and range of data sets. It explains that the median is the middle value of a data set in numerical order, or the mean of the two middle values for an even number of values. The mode is the value that occurs most frequently in a data set. The range is the difference between the greatest and least values. Examples are given of calculating each for a set of monthly temperature data.
Alg 2 section 1.6
The document discusses solving linear inequalities and includes three topics: forms of linear inequalities, compound inequalities, and homework assigned which includes odd numbered problems 23 through 33 and 37 through 47 on page 45 of the text.
Pc sections p.6-p.7
This document lists key properties of numbers and operations covered in two sections. Section P.6 covers commutative and associative properties of addition and multiplication as well as the distributive property of multiplication over addition and subtraction. Section P.7 directs students to complete specific math problems from two different pages focusing on odd numbered questions.
Alg 1 sections 1.5-1.6
1. This document discusses different ways to represent functions including equations, tables, and mapping diagrams. It provides examples of each.
2. Strategies for solving problems are outlined, including using formulas, patterns, and drawing diagrams. Specific math problems are provided as examples.
3. Homework assignments are listed at the end, including problem numbers from pages 31-32 and 38-39.
Alg2 sections 1.4-1.5
This document provides an overview of rewriting formulas and equations. It includes sections on common formulas, solving for variables, examples to practice, and application problems. Formulas covered include distance, temperature, areas of shapes, and circumference. It also discusses strategies for problem solving like using formulas, patterns, and diagrams. Sample problems are provided for average speed, finding heights from a pattern, and positioning posters on a wall. The document concludes with assigned homework problems.
Pc sections p.4-p.5
This document provides an overview of topics in algebra including:
1) Different forms of equations for lines (slope-intercept, point-slope, etc.) and quadratics (general, vertex).
2) How to determine if lines are parallel or perpendicular.
3) Methods for graphing and solving linear and quadratic equations (finding intercepts, using a graphing calculator, extracting roots, using the quadratic formula).
4) An example word problem on investing money with different interest rates and determining the amount to invest to earn a target return.
5) Assigned homework problems from the textbook.
Pa sections 1-2
This document provides instructions for performing common mathematical operations like addition, subtraction, multiplication, and division. It includes examples of writing down a number and then performing sequential operations on it like doubling, adding or subtracting other numbers, dividing, and subtracting the original number. The document also lists common words used to indicate different operations and directs the reader to specific math problems to complete on designated pages.
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Lesson 7.1
- 1. Chapter 7 Linear Systems •A system of equations is a set of two or more equations that use the same variables. 2 x − y + z = 4 A brace is y = x + 3 x + 3 y − z = 11 used to keep 4 x + y − z = 14 y = −2 x + 3 the equations of a system together. •If the graph of each equation in a system of two variables is a line then the system is a linear system.
- 2. Chapter 7 Linear Systems They can Coinciding Parallel lines intersect. lines, the same line. One solution Infinite solution No solution. Consistent Consistent Inconsistent Independent Dependent