# 3 2: Solving Systems in Three Variables

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 3 2: Solving Systems in Three Variables What is a system of 3 Equations with 3 Unknowns? A system of 3 equations with 3 unknowns is a list of three equations that have 3 variables in common. Each equation can represent a plane in space. It is common to use the variables x, y, and z as the three variables to represent a plane (flat surface) in a three dimensional space. Applications in science and business use other variables that better reflect the given application. They reflect the relationship between the objects the variables represent. If a hospital wants to represent the relationship between the number of doctors, nurses and patients in three different wards they will create three equations that each use D, N and P as the variables. The variables can be any letters. This section will choose to use x, y and w as the 3 variables for all the problems in this unit. Examples of a System of 3 Equations with 3 Unknowns Equation A 2x 3y + 4w = 9 5x + 2y 9w = 2 x + y w = 3 Equation A 2x + 4w = 9 5x + 2y = 2 y w = 3 What is a solution to a System of Three Equations with Three Unknowns? The solution to a system of three equations with three unknowns is an ordered triplet of numbers. The order for each ordered triplet of numbers will be (x, y, w). ( 3, 5, 2 ) 3, 0, 1 2 3, 2 5, 3 4 This section will choose to use x, y and w as the 3 variables for all the problems in this unit. The solution to a system of three equations with three unknowns is an ordered triplet of numbers. The order for each ordered triplet of numbers will be (x, y, w). An ordered triplet is a solution to a system of three equations if each equation is true when you substitute the values for each variable into all three equation. To determine if an ordered triplet is a solution the values for x y, and z will be plugged into all 3 equations for the x, y and z variables. If the ordered triplet (x, y, w) make all 3 equations true then the ordered triplet (x, y, w) is a solution. If the ordered triplet (x, y, w) does not make all 3 equations true then it is not a solution. Math 120 Section 3 2 Page Eitel

2 Is (x, y, w) a solution to the system? Is ( 1,2,3) a solution to Example 1 Equation A 2x 3y + w = 1 5x + 2y w = 6 x + y 2w = 3 ( 1,2,3) means x =1, y = 2 and w = 3 plug these values into both Equation A, and to see if they work in all 3 equations x =1, y = 2 and w = 3 Equation A 2x 3y + w = 1 5x + 2y w = 6 x + y 2w = 3 2(1) 3(2) + (3) = 1 5(1) + 2(2) (3) = 6 (1) + (2) 2(3) = 3 ( 1,2,3) works in all 3 equations so YES it is a solution. Is ( 3,1, 1 ) a solution to Example 2 Equation A 2x + 4w = 1 5x + 2y =17 y w = 2 ( 3,1, 1 ) means x = 3, y =1 and w = 1 plug these values into both Equation A, and to see if they work in all 3 equations Equation A 2x + 4w = 1 5x + 2y =17 y w = 2 ( 3,1, 1 ) works in all 3 equations so YES it is a solution. x = 3, y =1 and w = 1 2(3) + 4( 1) = 1 5(3) + 2(2) =17 (1) 1( 1) = 2 Math 120 Section 3 2 Page Eitel

3 Is (x, y, w) a solution to the system? Is ( 2,0, 1) a solution to Example 3 Equation A x 4y + w =1 2x + 3y w = 5 x + y = 3 ( 2,0, 1) means x = 2, y = 0 and w = 1 plug these values into both Equation A, and to see if they work in all 3 equations x = 2, y = 0 and w = 1 Equation A x 4y + w =1 2x + 3y w = 5 x + y = 3 (2) 4(0) + ( 1) =1 2(2) + 3(0) ( 1) = 5 (2) + (0) 3 ( 2,0, 1) does NOT work in equation C No it is NOT a solution. Math 120 Section 3 2 Page Eitel

4 Solving Systems of 3 Equations with 3 Unknowns The general procedure to solve a system of three equations with three unknowns is to use 2 elimination steps to reduce the problem to a system of two equations with two unknowns. You then proceed to solve the system of two equations with two unknowns just like you did in the previous section. Steps for Solving Systems of 3 Equations with 3 Unknowns Equation A 2x 3y + 3w =12 2x + 2y w = 6 2x + y w = 4 Step 1. Use a combination of Equation A and to eliminate one of the three variables. This will give you a new equation with two variables. Call this new equation Equation D Step 2. Use a combination of and to eliminate the same variable as that was eliminated in step 1. This will give you a new equation with two variables. Call this new equation Equation E You will now have 2 new equations. and Equation D. Each of the new equations will have the same two variables. Step 3: Use a combination of Equation D and Equation E to eliminate one of the two varaibles. This will allow you to solve for the variable that remains after elimination. Step 4: Substitute the value of the variable that was solved for in step 3 into Equation D and solve for the variable that remains. Step 5: Substitute the value of the variables that were solved for in step 3 and step 4 into Equation A and solve for the remaining variable. You now have the values for x, y and w. State the answer as a ordered triplet. Math 120 Section 3 2 Page Eitel

5 Example 1 Equation A 2x 3y + 3w =1 2x + 2y w = 3 x + y w = 4 Reduce the system with Three Unknowns to a new System with 2 Unknowns We will choose to eliminate the x terms. Step 1: Step 2: Add Equation A and Add and 2 times to eliminate the x terms to eliminate the x terms A+ B will eliminate the x terms B+ 2 C will eliminate the x terms Equation A 2x 3y + 3w =1 2x + 2y w = 3 2x + 2y w = 3 2 2x + 2y 2w = 8 y + 2w = 2 4y 3w = 5 call the new equation Equation D call the new equation Equation E New System with 2 Unknowns Equation D y + 2w = 2 Equation E 4 y 3w = 8 Math 120 Section 3 2 Page Eitel

6 Step 3: Solve the System with 2 Unknowns by Elimination We will choose to eliminate the y terms and solve for w. Equation D y + 2w = 2 Equation E 4 y 3w = 8 4 Equation D + Equation E will eliminate the y terms 4 Equation D 4y + 8w = 8 Equation E 4y 3w = 8 2y = 4 Solve for y y = 2 Step 3: Plug y = 2 into Equation D: y + 2w = 6 and solve for w (2)+ 2w = 6 2w = 8 w = 4 Step 4: Plug y = 2 and w = 4 into Equation A: 2x 3y + 3w =12 and solve for x 2x 3(2) + 3(4) =12 2x = 6 x = 3 Answer: (3,2,1) Step 5: Check: Equation A 2x 3y + 3w =12 2x + 2y w = 6 2x + y w = 4 2(3) 3(2) + 3(4) =12 2(3) + 2(2) (4) = 6 2(3) + (2) (4) = 4 Math 120 Section 3 2 Page Eitel

7 Equation A Solving Systems with 3 Unknowns Example 2 2x + 3y + 2w = 9 x 3y w = 4 x + y w = 0 Step 1: Reduce the system with Three Unknowns to a new System with 2 Unknowns We will choose to eliminate the y terms. Add Equation A and to eliminate the y terms call the new equation Equation D Equation A 2x + 3y + 2w = 9 x 3y w = 4 3x + w = 5 Add and 3 to eliminate the y terms call the new equation Equation E x 3y w = 4 3 3x + 3y 3w = 0 4x 4w = 4 New System with 2 Unknowns Equation D 3x + w = 5 Equation E 4 x 4w = 4 Math 120 Section 3 2 Page Eitel

8 Step 2: Solve the System with 2 Unknowns by Elimination We will choose to eliminate the w terms and solve for y. Equation D 3x + w = 5 Equation E 4 x 4w = 4 Add 4 Equation D and Equation E to eliminate the w terms 4 Equation D 12x + 4w = 20 Equation E 4x 4w = 4 16x =16 Solve for 1 x =1 Step 3: Plug x =1 into Equation D: 3x + w = 5 and solve for w 3(1)+ w = 5 3+ w = 5 w = 2 Step 4: Plug x =1 and w = 2 into Equation A: 2x + 3y + 2w = 9 and solve for y 2(1) + 3y + 2(2) = 9 3y = 3 y =1 Answer: (1,1,2) Step 5: Check: Equation A 2x + 3y + 2w = 9 x 3y w = 4 x + y w = 0 2(1) + 3(1) + 2(2) = 9 (1) 3(1) (2) = 4 (1) + (1) (2) = 0 Math 120 Section 3 2 Page Eitel

9 Equation A Solving Systems with 3 Unknowns Example 3 3x y = 7 3x 2y + 2w = 4 2x 3y + w = 5 Step 1: Reduce the system with Three Unknowns to a new System with 2 Unknowns We will choose to eliminate the w terms since Equation A has no w terms. Equation A has no w terms so we use it as is and call it equation Equation D Add and 2 to eliminate the w terms call the new equation Equation E Equation A: 3x y = 7 3x 2y + 2w = 4 2 4x + 6y 2w =10 x + 4y = 6 New System with 2 Unknowns Equation D 3x y = 7 Equation E x + 4y = 6 Math 120 Section 3 2 Page Eitel

10 Step 2: Solve the New System with 2 Unknowns by Elimination We will choose to eliminate the y terms and solve for x. Equation D 3x y = 7 Equation E x + 4y = 6 Add 4 Equation D and Equation E to eliminate the w terms 4 Equation D 12x y = 28 Equation E x + 4 y = 6 11x = 22 Solve for x x = 2 Step 3: Plug x = 2 into Equation D: 3x y = 7 and solve for y 3( 2) y = 7 y = 1 y =1 Step 4: Plug x = 2 and y =1 into : 3x 2y + 2w = 4 and solve for w 3( 2) 2(1) + 2w = 4 2w = 4 w = 2 Answer: ( 2,1,2) Step 5: Check: Equation A 3x y = 7 3x 2y + 2w = 4 2x 3y + w = 5 3( 2) (1) = 7 3( 2) 2(1) + 2(2) = 4 2( 2) 3(1) + (2) = 5 Math 120 Section 3 2 Page Eitel

11 The Intersection of a system of two equations with two variables The Intersection of Two Lines on a Plane has 3 Possible Cases Two linear equations with two variables x and y can be modeled as the equations of two lines on a plane. There are three different geometric descriptions for the three possible ways that two lines on a plane can meet. Each different geometric description has a different type of solution for the system of two linear equations. 1. If the two lines intersect then they have a single point in common. That single point is the only ordered pair (x,y) that makes both of the equations true. The only solution to this system is the ordered pair (x, y) where the two lines intercept. 2. If the two lines have the same slope but a different y intercepts then the two lines are parallel. Parallel lines do not intersect so they have no points in common. A system of two parallel lines have no solution. 3. If one equation for the line are a multiple of the other equation then the two lines have the same slope but a different y intercept. Both equations represent the same line. All the points on one line are also on the other line because the equations represent the same line. Either linear equation can represent the solution to the system as they are the same equation but in a different form. Math 120 Section 3 2 Page Eitel

12 The Intersection of a system of three equation with three variables. A similar geometric description apples to a system of three equation with three variables. An equation with the variables (x, y, z ) can be modeled as a plane or a flat surface that extends without end. A system of three equation with three variables represents three planes. There are three general statements that describe possible arraignments of three planes in space. The three planes can meet at a single point. The three planes can have no points in common to all three planes. The three planes can meet at an infinite number of points. The geometric drawings of these types of systems is not as simple as is was with two lines. The only way two lines on a plane can have no points in common is if the two lines are parallel. This concept cannot be extended to three planes in space. The three planes may all be may be parallel to each other but other graphical arraignments are also possible that produce three planes that do not have a point common to all three planes. It can be difficult to determine the exact arraignment of the three planes without graphing them. Graphing three planes in a three dimensional system by hand can be difficult, It is common to use a computer software program to do this. The Intersection of Three Planes in Space has Several Possible Cases 1. Three planes can intersect so that they have a one point in common. That single point in space is the only ordered triplet (x, y, z) that makes all three of the equations true. This unit will only deal with the case where the three planes meet at one common point. Future courses will develop the solutions to these other cases in detail. 2. Three planes can be placed so that they three different planes that are parallel to each other. Parallel planes do not intersect so they have no points in common. A system of parallel planes have no solution. 3. Three planes can be placed so that any two of the three planes intersect each other but all three planes do not intersect each other in a common point. The planes do not have a point common to all three planes. Such a system of three planes have no solution. 4. Three planes can be placed so that all three planes intersect at a common line. Any point (x, y, z ) on the line of intersection will make al three equations true. The linear equation of the intersection line represents the solution to the system. 5. If each equation for the three planes are a multiple of the other equation then the three equations represent the same plane. All the points on one plane are also on the other plane. All the points on one plane are also on the other planes because the other equations represent the same plane. Any of the three equations can represent the solution to the system as they are the same equation but in a different form. Math 120 Section 3 2 Page Eitel

### 5 Systems of Equations

Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

### Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

### 3 1B: Solving a System of Equations by Substitution

3 1B: Solving a System of Equations by Substitution 1. Look for an equation that has been solved for a single variable. Lets say 2. Substitute the value of that variable from into in place of that variable.

### MathQuest: Linear Algebra. 1. What is the solution to the following system of equations? 2x+y = 3 3x y = 7

MathQuest: Linear Algebra Systems of Equations 1. What is the solution to the following system of equations? 2x+y = 3 3x y = 7 (a) x = 4 and y = 5 (b) x = 4 and y = 5 (c) x = 2 and y = 1 (d) x = 2 and

### Lesson 22: Solution Sets to Simultaneous Equations

Student Outcomes Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically. Classwork Opening Exercise

### Solving Equations Involving Parallel and Perpendicular Lines Examples

Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines

### SYSTEMS OF LINEAR EQUATIONS

SYSTEMS OF LINEAR EQUATIONS Sstems of linear equations refer to a set of two or more linear equations used to find the value of the unknown variables. If the set of linear equations consist of two equations

### Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

### Chapter 8 Graphs and Functions:

Chapter 8 Graphs and Functions: Cartesian axes, coordinates and points 8.1 Pictorially we plot points and graphs in a plane (flat space) using a set of Cartesian axes traditionally called the x and y axes

### Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown.

Warm Up Write an equation given the slope and y-intercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and y-intercept From the graph, you can see that the slope is

### Reasoning with Equations and Inequalities

Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations.

### 10.1 Systems of Linear Equations: Substitution and Elimination

726 CHAPTER 10 Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Sustitution and Elimination PREPARING FOR THIS SECTION Before getting started, review the following: Linear Equations

### The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6

Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =

Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the

### Slope-Intercept Equation. Example

1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine

### Intersecting Two Lines, Part One

Module 1.4 Page 97 of 938. Module 1.4: Intersecting Two Lines, Part One This module will explain to you several common methods used for intersecting two lines. By this, we mean finding the point x, y)

### Systems of Linear Equations and Inequalities

Systems of Linear Equations and Inequalities Recall that every linear equation in two variables can be identified with a line. When we group two such equations together, we know from geometry what can

### 3.1 Solving Systems Using Tables and Graphs

Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### 2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

### Systems of Linear Equations in Three Variables

5.3 Systems of Linear Equations in Three Variables 5.3 OBJECTIVES 1. Find ordered triples associated with three equations 2. Solve a system by the addition method 3. Interpret a solution graphically 4.

### What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### The Graph of a Linear Equation

4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

### The Point-Slope Form

7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

### Let s explore the content and skills assessed by Heart of Algebra questions.

Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,

### Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t

Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are

### Make sure you look at the reminders or examples before each set of problems to jog your memory! Solve

Name Date Make sure you look at the reminders or examples before each set of problems to jog your memory! I. Solving Linear Equations 1. Eliminate parentheses. Combine like terms 3. Eliminate terms by

### Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

### Write the Equation of the Line Review

Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections

### SPIRIT 2.0 Lesson: A Point Of Intersection

SPIRIT 2.0 Lesson: A Point Of Intersection ================================Lesson Header============================= Lesson Title: A Point of Intersection Draft Date: 6/17/08 1st Author (Writer): Jenn

### Math 113 Review for Exam I

Math 113 Review for Exam I Section 1.1 Cartesian Coordinate System, Slope, & Equation of a Line (1.) Rectangular or Cartesian Coordinate System You should be able to label the quadrants in the rectangular

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

### Learning Goals. b. Calculate the total cost of using Roller Rama. y 5 5(50) 5 250. The total cost of using Roller Rama is \$250.

Making Decisions Using the Best Method to Solve a Linear System Learning Goals In this lesson, you will: Write a linear system of equations to represent a problem context. Choose the best method to solve

### Solving Systems of Two Equations Algebraically

8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### 1 Functions, Graphs and Limits

1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its

### Graphing Equations. with Color Activity

Graphing Equations with Color Activity Students must re-write equations into slope intercept form and then graph them on a coordinate plane. 2011 Lindsay Perro Name Date Between The Lines Re-write each

### GRAPHING LINEAR EQUATIONS IN TWO VARIABLES

GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: Slope-Intercept Form: y = mx+ b In an equation

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### 12.5 Equations of Lines and Planes

Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P

### Instructor: Laura Dostert Course: Summer Math Refresher

Student: Date: Instructor: Laura Dostert Course: Summer Math Refresher Assignment: Review 8 Equations of Lines 1. Using the slope formula, find the slope of the line through the given points. (8,0) and

### Economics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013

Economics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013 Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on

### Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

### Lesson 9: Graphing Standard Form Equations Lesson 2 of 2. Example 1

Lesson 9: Graphing Standard Form Equations Lesson 2 of 2 Method 2: Rewriting the equation in slope intercept form Use the same strategies that were used for solving equations: 1. 2. Your goal is to solve

### Section 8.4 - Composite and Inverse Functions

Math 127 - Section 8.4 - Page 1 Section 8.4 - Composite and Inverse Functions I. Composition of Functions A. If f and g are functions, then the composite function of f and g (written f g) is: (f g)( =

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### Georgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade

Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade

### Section 1.10 Lines. The Slope of a Line

Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes through

### High School Algebra Reasoning with Equations and Inequalities Solve systems of equations.

Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student

### GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### 3-4 Equations of Lines

3-4 Equations of Lines Write an equation in slope-intercept form of the line having the given slope and y-intercept. Then graph the 1.m: 4, y-intercept: 3 2. y-intercept: 1 The slope-intercept form of

### Section 1.5 Linear Models

Section 1.5 Linear Models Some real-life problems can be modeled using linear equations. Now that we know how to find the slope of a line, the equation of a line, and the point of intersection of two lines,

### Questions. Strategies August/September Number Theory. What is meant by a number being evenly divisible by another number?

Content Skills Essential August/September Number Theory Identify factors List multiples of whole numbers Classify prime and composite numbers Analyze the rules of divisibility What is meant by a number

### Lecture 1: Systems of Linear Equations

MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

### 2.3 Writing Equations of Lines

. Writing Equations of Lines In this section ou will learn to use point-slope form to write an equation of a line use slope-intercept form to write an equation of a line graph linear equations using the

Task Model 3 Equation/Numeric DOK Level 1 algebraically, example, have no solution because 6. 3. The student estimates solutions by graphing systems of two linear equations in two variables. Prompt Features:

### Slope-Intercept Form of a Linear Equation Examples

Slope-Intercept Form of a Linear Equation Examples. In the figure at the right, AB passes through points A(0, b) and B(x, y). Notice that b is the y-intercept of AB. Suppose you want to find an equation

### Linear Equations ! 25 30 35\$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development

MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!

### High School Functions Interpreting Functions Understand the concept of a function and use function notation.

Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra.

### Question 2: How will changes in the objective function s coefficients change the optimal solution?

Question 2: How will changes in the objective function s coefficients change the optimal solution? In the previous question, we examined how changing the constants in the constraints changed the optimal

### IV. ALGEBRAIC CONCEPTS

IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other

### Graphing Linear Equations in Slope-Intercept Form

4.4. Graphing Linear Equations in Slope-Intercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope

### Systems of Linear Equations

DETAILED SOLUTIONS AND CONCEPTS - SYSTEMS OF LINEAR EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE

### Pre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems

Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small

### Writing the Equation of a Line in Slope-Intercept Form

Writing the Equation of a Line in Slope-Intercept Form Slope-Intercept Form y = mx + b Example 1: Give the equation of the line in slope-intercept form a. With y-intercept (0, 2) and slope -9 b. Passing

### Graphing Linear Equations

Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### 5. Equations of Lines: slope intercept & point slope

5. Equations of Lines: slope intercept & point slope Slope of the line m rise run Slope-Intercept Form m + b m is slope; b is -intercept Point-Slope Form m( + or m( Slope of parallel lines m m (slopes

### with functions, expressions and equations which follow in units 3 and 4.

Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model

### LINEAR EQUATIONS IN TWO VARIABLES

66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

### c sigma & CEMTL

c sigma & CEMTL Foreword The Regional Centre for Excellence in Mathematics Teaching and Learning (CEMTL) is collaboration between the Shannon Consortium Partners: University of Limerick, Institute of Technology,

### Solving Systems of Linear Equations Graphing

Solving Systems of Linear Equations Graphing Outcome (learning objective) Students will accurately solve a system of equations by graphing. Student/Class Goal Students thinking about continuing their academic

### The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

### Algebra 1 Topic 8: Solving linear equations and inequalities Student Activity Sheet 1; use with Overview

Algebra 1 Topic 8: Student Activity Sheet 1; use with Overview 1. A car rental company charges \$29.95 plus 16 cents per mile for each mile driven. The cost in dollars of renting a car, r, is a function

### Multiplying Polynomials 5

Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive

### The Cartesian Plane The Cartesian Plane. Performance Criteria 3. Pre-Test 5. Coordinates 7. Graphs of linear functions 9. The gradient of a line 13

6 The Cartesian Plane The Cartesian Plane Performance Criteria 3 Pre-Test 5 Coordinates 7 Graphs of linear functions 9 The gradient of a line 13 Linear equations 19 Empirical Data 24 Lines of best fit

### A floor is a flat surface that extends in all directions. So, it models a plane. 1-1 Points, Lines, and Planes

1-1 Points, Lines, and Planes Use the figure to name each of the following. 1. a line containing point X 5. a floor A floor is a flat surface that extends in all directions. So, it models a plane. Draw

### Sect The Slope-Intercept Form

Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### Algebra 2 PreAP. Name Period

Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

### 1. Graphing Linear Inequalities

Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means

### Math 215 HW #1 Solutions

Math 25 HW # Solutions. Problem.2.3. Describe the intersection of the three planes u+v+w+z = 6 and u+w+z = 4 and u + w = 2 (all in four-dimensional space). Is it a line or a point or an empty set? What

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### Section 12.6: Directional Derivatives and the Gradient Vector

Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

### Course Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)

Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,

### Algebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2

1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3-dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect

### Foundations of Mathematics 11 (Online)

Course Outline Coquitlam Learning Opportunity Centre 104-2748 Lougheed Hwy Port Coquitlam, BC, V3B 6P2 Phone: (604) 945-4211 Course Name Teacher Course Format Teacher Contact Information & Schedule Learning

### How can you write an equation of a line when you are given the slope and the y-intercept of the line? ACTIVITY: Writing Equations of Lines

. Writing Equations in Slope-Intercept Form How can ou write an equation of a line when ou are given the slope and the -intercept of the line? ACTIVITY: Writing Equations of Lines Work with a partner.

### LINEAR FUNCTIONS OF 2 VARIABLES

CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for

### Alex and Morgan were asked to graph the equation y = 2x + 1

Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

### Correlation key concepts:

CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)

### Systems of Equations - Substitution

4.2 Systems of Equations - Substitution Objective: Solve systems of equations using substitution. When solving a system by graphing has several limitations. First, it requires the graph to be perfectly