# Solving Systems of Linear Equations by Elimination

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 5.3 Solving Systems of Linear Equations by Elimination How can you use elimination to solve a system of linear equations? ACTIVITY: Using Elimination to Solve a System Work with a partner. Solve each system of linear equations by using two methods. Method : Subtract. Subtract Equation from Equation. What is the result? Explain how you can use the result to solve the system of equations. Method : Add. Add the two equations. What is the result? Explain how you can use the result to solve the system of equations. Is the solution the same using both methods? a. x + y = 4 b. 3x y = 4 c. x + y = 7 x y = 0 3x + y = x y = 5 ACTIVITY: Using Elimination to Solve a System Work with a partner. COMMON CORE Systems of Equations In this lesson, you will write and solve systems of linear equations by elimination. solve real-life problems. Learning Standards 8.EE.8b 8.EE.8c x + y = Equation x + 5y = Equation a. Can you add or subtract the equations to solve the system of linear equations? Explain. b. Explain what property you can apply to Equation in the system so that the y-coefficients are the same. c. Explain what property you can apply to Equation in the system so that the x-coefficients are the same. d. You solve the system in part (b). Your partner solves the system in part (c). Compare your solutions. e. Use a graphing calculator to check your solution. 6 Chapter 5 Systems of Linear Equations

2 3 ACTIVITY: Solving a Secret Code Math Practice Find Entry Points What is the first thing you do to solve a system of linear equations? Why? Work with a partner. Solve the puzzle to find the name of a famous mathematician who lived in Egypt around 350 A.D B W R M F Y K N O J A S I D X Z Q P C E G B T J M R C Z N O U W K X U H L Y S Q F E A S W K R M G J Z N H V D G 3 E L X L F Q O B x y x y 4 x y 3 x y 0 x y 4 x y x y 5 x y 5 x y 0 x y 3 3x 3y 0 x y 8 x y 5 x y 4. IN YOUR OWN WORDS How can you use elimination to solve a system of linear equations? 5. STRUCTURE When can you add or subtract equations in a system to solve the system? When do you have to multiply first? Justify your answers with examples. 6. LOGIC In Activity, why can you multiply equations in the system by a constant and not change the solution of the system? Explain your reasoning. Use what you learned about systems of linear equations to complete Exercises 4 6 on page. Section 5.3 Solving Systems of Linear Equations by Elimination 7

3 5.3 Lesson Lesson Tutorials Solving a System of Linear Equations by Elimination Step : Multiply, if necessary, one or both equations by a constant so at least pair of like terms has the same or opposite coefficients. Step : Add or subtract the equations to eliminate one of the variables. Step 3: Solve the resulting equation for the remaining variable. Step 4: Substitute the value from Step 3 into one of the original equations and solve. Study Tip Check Equation x + 3y = 7 + 3( 3) =? = Equation EXAMPLE Because the coefficients of x are the same, you can also solve the system by subtracting in Step. x + 3y = x 3y = 6 6y = 8 So, y = 3. x 3y = 6 7 3( 3) =? 6 6 = 6 Solving a System of Linear Equations by Elimination Solve the system by elimination. x + 3y = Equation x 3y = 6 Equation Step : The coefficients of the y-terms are already opposites. Step : Add the equations. x + 3y = Equation x 3y = 6 Equation x = 4 Add the equations. Step 3: Solve for x. x = 4 Equation from Step x = 7 Divide each side by. Step 4: Substitute 7 for x in one of the original equations and solve for y. x + 3y = Equation 7 + 3y = Substitute 7 for x. 3y = 9 Subtract 7 from each side. y = 3 Divide each side by 3. The solution is (7, 3). Exercises 7 Solve the system of linear equations by elimination. Check your solution.. x y = 9. 5x + y = x + 4y = 6 4x + y = 5x + y = 7x + 4y = 4 8 Chapter 5 Systems of Linear Equations

4 EXAMPLE Solving a System of Linear Equations by Elimination Solve the system by elimination. 6x + 5y = 5 Equation x 4y = 4 Equation Step : Multiply Equation by 3. 6x + 5y = 5 6x + 5y = 5 Equation x 4y = 4 Multiply by 3. 6x y = 4 Revised Equation Study Tip In Example, notice that you can also multiply Equation by 3 and then add the equations. Step : Subtract the equations. 6x + 5y = 5 Equation 6x y = 4 Revised Equation Step 3: Solve for y. 7y = 7 Subtract the equations. 7y = 7 Equation from Step y = Divide each side by 7. Step 4: Substitute for y in one of the original equations and solve for x. x 4y = 4 Equation x 4( ) = 4 Substitute for y. x + 4 = 4 x = 0 Multiply. Subtract 4 from each side. x = 5 Divide each side by. The solution is ( 5, ). Check 0 6x 5y 5 x 4y Solve the system of linear equations by elimination. Check your solution. 4. 3x + y = 5. 4x 5y = y = 5 5x 6x + 3y = 4 x y = 8 y = x + 3 Section 5.3 Solving Systems of Linear Equations by Elimination 9

5 EXAMPLE 3 Real-Life Application You buy 8 hostas and 5 daylilies for \$93. Your friend buys 3 hostas and daylilies for \$7. Write and solve a system of linear equations to find the cost of each daylily. Use a verbal model to write a system of linear equations. Number of hostas Cost of each hosta, x + Number of daylilies Cost of each daylily, y = Total cost The system is: 8x + 5y = 93 Equation (You) 3x + y = 7 Equation (Your friend) Step : To find the cost y of each daylily, eliminate the x-terms. Multiply Equation by 3. Multiply Equation by 8. 8x + 5y = 93 Multiply by 3. 4x + 45y = 579 Revised Equation 3x + y = 7 Multiply by 8. 4x + 96y = 936 Revised Equation Step : Subtract the revised equations. 4x + 45y = 579 Revised Equation 4x + 96y = 936 Revised Equation 5y = 357 Subtract the equations. Step 3: Solving the equation 5y = 357 gives y = 7. So, each daylily costs \$7. Exercises 6 7. A landscaper buys 4 peonies and 9 geraniums for \$90. Another landscaper buys 5 peonies and 6 geraniums for \$85. Write and solve a system of linear equations to find the cost of each peony. Methods for Solving Systems of Linear Equations Method Graphing (Lesson 5.) Substitution (Lesson 5.) Elimination (Lesson 5.3) Elimination (Multiply First) (Lesson 5.3) To estimate solutions When to Use When one of the variables in one of the equations has a coefficient of or When at least pair of like terms has the same or opposite coefficients When one of the variables cannot be eliminated by adding or subtracting the equations 0 Chapter 5 Systems of Linear Equations

6 5.3 Exercises Help with Homework. WRITING Describe how to solve a system of linear equations by elimination.. NUMBER SENSE When should you use multiplication to solve a system of linear equations by elimination? 3. WHICH ONE DOESN T BELONG? Which system of equations does not belong with the other three? Explain your reasoning. 3x + 3y = 3 x 3y = 7 x + y = 6 x 3y = 0 x + 3y = 3x y = 0 x + y = 5 3x y = 3 9+(-6)=3 3+(-3)= 4+(-9)= 9+(-)= Use a method from Activity to solve the system. 4. x + y = 3 5. x + 3y = x + y = 3 x y = x + 3y = 3x y = 9 Solve the system of linear equations by elimination. Check your solution. 7. x + 3y = 5 8. x y = x + 3y = 5 x y = 3 3x + y = 3 x + 3y = 0 0. x + 7y =. x + 5y = 6. 3x y = 4 x 4y = 3x 5y = 6x y = 3. ERROR ANALYSIS Describe and correct the error in solving the system of linear equations. 4. RAFFLE TICKETS You and your The solution is (5, 8). friend are selling raffle tickets for a new laptop. You sell 4 more tickets than your friend sells. Together, you and your friend sell 58 tickets. a. Write a system of linear equations that represents this situation. b. How many tickets does each of you sell? 5. JOGGING You can jog around your block twice and the park once in 0 minutes. You can jog around your block twice and the park 3 times in minutes. a. Write a system of linear equations that represents this situation. b. How long does it take you to jog around the park? 5x + y = 9 Equation 3x y = Equation x = 0 x = 5 Section 5.3 Solving Systems of Linear Equations by Elimination

7 3 Solve the system of linear equations by elimination. Check your solution. 6. x y = 0 7. x + 4y = 8. x + 3y = 7 3x y = 3 3x + 5y = 0 5x + 8y = 9. 3x + 3 = 3y 0. x 6 = 4y. 5x = 4y + 8 x 6y = 7y = 3x + 9 3y = 3x 3. ERROR ANALYSIS Describe and correct the error in solving the system of linear equations. x + y = Equation Multiply by 5. 5x + 5y = 5 5x + 3y = 3 Equation 5x + 3y = 3 8y = 8 y = The solution is (, ). 3. REASONING For what values of a and b should you solve the system by elimination? a. 4x y = 3 b. x 7y = 6 ax + 0y = 6 6x + by = 9 Determine whether the line through the first pair of points intersects the line through the second pair of points. Explain. 4. Line : (, ), (, 7) 5. Line : (3, ), (7, ) Line : ( 4, ), (0, 5) Line : (5, ), (6, ) 6. AIRPLANES Two airplanes are flying to the same airport. Their positions are shown in the graph. Write a system of linear equations that represents this situation. Solve the system by elimination to justify your answer. y Airport x 7. TEST PRACTICE The table shows the number of correct answers on a practice standardized test. You score 86 points on the test, and your friend scores 76 points. You Your Friend Multiple Choice 3 8 Short Response 0 5 a. Write a system of linear equations that represents this situation. b. How many points is each type of question worth? Chapter 5 Systems of Linear Equations

8 8. LOGIC You solve a system of equations in which x represents the number of adult tickets sold and y represents the number of student tickets sold. Can ( 6, 4) be the solution of the system? Explain your reasoning. 9. VACATION The table shows the activities of two tourists at a vacation resort. You want to go parasailing for hour and horseback riding for hours. How much do you expect to pay? Parasailing Horseback Riding Total Cost Tourist hours 5 hours \$05 Tourist 3 hours 3 hours \$ REASONING The solution of a system of linear equations is (, 4). One equation in the system is x + y = 0. Explain how you could find a second equation for the system. Then find a second equation. Solve the system by elimination to justify your answer. 3. JEWELER A metal alloy is a mixture of two or more metals. A jeweler wants to make 8 grams of 8-carat gold, which is 75% gold. The jeweler has an alloy that is 90% gold and an alloy that is 50% gold. How much of each alloy should the jeweler use? 3. PROBLEM SOLVING A powerboat takes 30 minutes to travel 0 miles downstream. The return trip takes 50 minutes. What is the speed of the current? 33. Solve the system of equations by elimination. x y + 3z = x + y 4z = y z = 0 Decide whether the two equations are equivalent. (Section. and Section.3) 34. 4n + = n a + 6 = 36. 7v 3 = 5 3n = 9 a + 3 = 6 4v 3 = MULTIPLE CHOICE Which line has the same slope as y = x 3? (Section 4.4) A y = x + 4 B y = x + 3 C y x = 5 D y x = 7 Section 5.3 Solving Systems of Linear Equations by Elimination 3

### Solving Systems of Linear Equations by Substitution

4.2 Solving Systems of Linear Equations by Substitution How can you use substitution to solve a system of linear equations? 1 ACTIVITY: Using Substitution to Solve a System Work with a partner. Solve each

### 6-3 Solving Systems by Elimination

Warm Up Simplify each expression. 1. 2y 4x 2(4y 2x) 2. 5(x y) + 2x + 5y Write the least common multiple. 3. 3 and 6 4. 4 and 10 5. 6 and 8 Objectives Solve systems of linear equations in two variables

### Subtracting to Eliminate We first need to decide which variable is easiest to eliminate. Consider the following system of equations.

3.3 Solving Systems with Elimination Sometimes it is easier to eliminate a variable entirely from a system of equations rather than use the substitution method. We do this by adding opposite coefficients

### 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.

### 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

### Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent.

Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent. y = 2x 1 y = 2x + 3 The lines y = 2x 1 and y = 2x + 3 intersect at exactly one point

### Section 8.1 System of Linear Equations: Substitution and Elimination

Identifying Linear Systems Linear Systems in two variables Solution, 0 1 1 0 Consistent/ Independent Consistent/ Dependent Inconsistent Exactly one solution Infinitely many solutions No solution Two lines

### B. Examples - Use the given conditions to write a system of equations. Solve the system and answer the question.

Math 123 - Section 4.4 - Problem Solving Using Systems of Equations - Part II - Page 1 Section 4.4 Problem Solving Using Systems of Equations - Part II I. In general A. Remember to 1. Define the variables.

### Section 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a.

Algebra 1 Chapter 7 Notes Name Section 7.1 Solving Linear Systems by Graphing System of Linear Equations: Two or more equations in the same variables, also called a. Solution of a System of Linear Equations:

### Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:

Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system:

### Pre-AP Algebra 2 Lesson 2-1 Solving 2x2 Systems. 3x 2y 22 (6,2) 5x y 28 (1, 3)

Lesson 2-1 Solving 2x2 Systems Objectives: The students will be able to solve a 2 x 2 system of equations graphically and by substitution, as well as by elimination. Materials: paper, pencil, graphing

### Sect Solving Systems of Equations by Addition (Elimination)

Sect 4.3 - Solving Systems of Equations by Addition (Elimination) Concept # Solving System of Linear Equations in Two Variables by Addition (Elimination). A variant of the Addition Property of Equality

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### Systems of Linear Equations - Introduction

Systems of Linear Equations - Introduction What are Systems of Linear Equations Use an Example of a system of linear equations If we have two linear equations, y = x + 2 and y = 3x 6, can these two equations

### 5.1. Systems of Linear Equations. Linear Systems Substitution Method Elimination Method Special Systems

5.1 Systems of Linear Equations Linear Systems Substitution Method Elimination Method Special Systems 5.1-1 Linear Systems The possible graphs of a linear system in two unknowns are as follows. 1. The

### Study Guide and Review - Chapter 5

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. Set-builder notation is a less concise way of writing a solution set. The statement is false.

### Chapter 3: Systems of Linear Equalities and Inequalities

Chapter 3: Systems of Linear Equalities and Inequalities Chapter 3: Systems of Linear Equations and Inequalities Assignment Sheet Date Topic Assignment Completed 3.1 Solving Systems of Linear Equations

### Solving Systems of Linear Equations by Graphing

. Solving Sstems of Linear Equations b Graphing How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bed-and-breakfast. It

### Solving systems by elimination

December 1, 2008 Solving systems by elimination page 1 Solving systems by elimination Here is another method for solving a system of two equations. Sometimes this method is easier than either the graphing

### 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,

### Solving simultaneous equations. Jackie Nicholas

Mathematics Learning Centre Solving simultaneous equations Jackie Nicholas c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Simultaneous linear equations We will introduce

### 6-3 Elimination Using Addition and Subtraction. Use elimination to solve each system of equations. Multiply the second equation by 1.

Use elimination to solve each system of equations. 5m p = 7 7m p = 11 Multiply the second equation by 1. Then, add this to the first equation. Now, substitute 2 for m in either equation to find the value

### Equations of Lines Derivations

Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated

### Unit 6 Homework Key. Lesson 6.1. Graph the following linear equations using slope-intercept form. 1. 2 1 2. 3 4 3. 5 4. 2 1 5. 3 2 6. 5 7. 2 8. 1 9.

Lesson 6.1 Unit 6 Homework Key Graph the following linear equations using slope-intercept form. 1. 21 2. 34 3. 5 4. 21 5. 32 6. 5 7. 2 8. 1 9. 4 1 10. 22 11. 34 12. 45 13. 426 14. 639 15. 36 16. 2312 17.

### 3. Which of the following graphs shows the solution set for the inequality below? 2x + 4 > 2

1. Solve the following inequality. 24 < -2(x 3) < 36 A. -16 < x < -15 B. -21 < x < -9 C. -21 < x < -15 D. -15 < x < -9 2. Solve the following inequality. 3x + 4 < 8 A. x < B. < x < 4 C. -4 < x < D. -8

### 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

### Acquisition Lesson Plan for the Concept, Topic or Skill---Not for the Day

Acquisition Lesson Plan Concept: Linear Systems Author Name(s): High-School Delaware Math Cadre Committee Grade: Ninth Grade Time Frame: Two 45 minute periods Pre-requisite(s): Write algebraic expressions

### Solving Inequalities Using Addition or Subtraction 4.2. ACTIVITY: Writing an Inequality. ACTIVITY: Writing an Inequality

4.2 Solving Inequalities Using Addition or Subtraction solve an inequality? How can you use addition or subtraction to 1 ACTIVITY: Writing an Inequality Work with a partner. Members of the Boy Scouts must

### 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 Introduction

Solving Systems of Equations Introduction Outcome (learning objective) Students will write simple systems of equations and become familiar with systems of equations vocabulary terms. Student/Class Goal

### Unit 5: Analyze, Solve, and Graph Linear Inequalities

Unit 5 Table of Contents Unit 5: Analyze, Solve, and Graph Linear Inequalities Video Overview Learning Objectives 5.2 Media Run Times 5.3 Instructor Notes 5.4 The Mathematics of Linear Inequalities Writing,

### Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

### IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of \$2,000. She also earns \$500 for

### Multiple Representations of Equations & What We Know. Note that the worked out our turn and your turn charts can also be used as a matching activity.

Multiple Representations of s & What We California State Standards: 7 AF., 7 AF 3.3, 7 AF 3.4, Alg. 6., Alg. 7., Alg. 8. CCSS: 8.EE.6, 8.F., 8.F., 8.F.4 The idea of this lesson is to have students make

### 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

### Essential Question How can you describe the graph of the equation Ax + By = C? Number of adult tickets. adult

3. Graphing Linear Equations in Standard Form Essential Question How can ou describe the graph of the equation A + B = C? Using a Table to Plot Points Work with a partner. You sold a total of \$16 worth

### Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

### 21-114: Calculus for Architecture Homework #1 Solutions

21-114: Calculus for Architecture Homework #1 Solutions November 9, 2004 Mike Picollelli 1.1 #26. Find the domain of g(u) = u + 4 u. Solution: We solve this by considering the terms in the sum separately:

### 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

### Number of Solutions to Simultaneous Equations

Worksheet 3.5 Simultaneous Equations Section 1 Number of Solutions to Simultaneous Equations In maths we are sometimes confronted with two equations in two variables and we want to find out which values

### No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

### 7.1 LINEAR AND NONLINEAR SYSTEMS OF EQUATIONS

7.1 LINEAR AND NONLINEAR SYSTEMS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the method of substitution to solve systems of linear equations in two variables.

### Use the graph shown to determine whether each system is consistent or inconsistent and if it is independent or dependent.

Use the graph shown to determine whether each system is consistent or inconsistent and if it is independent or dependent. y = 3x + 1 y = 3x + 1 The two equations intersect at exactly one point, so they

### 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

### Algebra. Chapter 6: Systems of Equations and Inequalities. Name: Teacher: Pd:

Algebra Chapter 6: Systems of Equations and Inequalities Name: Teacher: Pd: Table of Contents Chapter 6-1: SWBAT: Identify solutions of systems of linear equations in two variables; Solve systems of linear

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

### Solving Special Systems of Linear Equations

5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest

### 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

### Chapter 4 SOLVING SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES

Chapter 4 SOLVING SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES 4.1 Introduction to Systems of Linear Equations: Solving by Graphing Objectives A Decide whether an ordered pair is a solution of a system

### Systems of Equations and Inequalities

Systems of Equations and Inequalities 6A Systems of Linear Equations Lab Solve Linear Equations by Using a Spreadsheet 6-1 Solving Systems by Graphing Lab Model Systems of Linear Equations 6-2 Solving

### Practice: Solving Systems of Equations (3 Different Methods)

-1- h AKEuotma ds7oofttlwoasrsew klglvch. Q laylly nroirgxhjtpsk rmeisqevravie9d.i U JMzaGdVex gw5ictuh7 iintfsi7n7iltler 9A7lmgLevb1rIai o1c.l Worksheet by Kuta Software LLC Da Vinci Design / Math 9 Practice:

### 3-1 Solving Systems of Equations. Solve each system of equations by using a table. 2. SOLUTION: Write each equation in slope-intercept form.

Solve each system of equations by using a table. 2. Write each equation in slope-intercept form. Also: Make a table to find a solution that satisfies both equations. Therefore, the solution of the system

### Solving Multi-Step Equations 1.2. ACTIVITY: Solving for the Angles of a Triangle

. Solving Multi-Step Equations How can you solve a multi-step equation? How can you check the reasonableness of your solution? ACTIVITY: Solving for the Angles of a Triangle Work with a partner. Write

### Chapter Exam Review for MAT098 - Prealgebra Chapters 1-2: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations

Chapter Exam Review for MAT098 - Prealgebra Chapters 1-2: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations Chapters 1-2 Learning Objectives: In chapter 1 students

### Solving and Graphing Linear Inequalities Chapter Questions. 6. How do graphs of linear inequalities compare with graphs of equations?

Solving and Graphing Linear Inequalities Chapter Questions 1. How do we translate a statement into an inequality? 2. Explain the steps to graphing an inequality on a number line. 3. How is solving an inequality

### 2-2 Solving One-Step Equations. Solve each equation. Check your solution. g + 5 = 33. Check: 104 = y. Check: Check: Page 1

Solve each equation. Check your solution. g + 5 = 33 104 = y 67 4 +Manual t = 7- Powered by Cognero esolutions Page 1 4+t= 7 a + 26 = 35 6 + c = 32 1.5 = y ( 5.6) Page 2 1.5 = y ( 5.6) Page 3 Page 4 Page

### Evaluate and Simplify Algebraic Expressions

1.2 Evaluate and Simplify Algebraic Expressions Before You studied properties of real numbers. Now You will evaluate and simplify expressions involving real numbers. Why? So you can estimate calorie use,

### Systems of Linear Equations: Elimination by Addition

OpenStax-CNX module: m21986 1 Systems of Linear Equations: Elimination by Addition Wade Ellis Denny Burzynski This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License

### ACTIVITY: Multiplying Binomials Using Algebra Tiles. Work with a partner. Six different algebra tiles are shown below.

7.3 Multiplying Polynomials How can you multiply two binomials? 1 ACTIVITY: Multiplying Binomials Using Algebra Tiles Work with a partner. Six different algebra tiles are shown below. 1 1 x x x x Write

### Apply and extend previous understandings of arithmetic to algebraic expressions. Draft

6th Grade - Expressions and Equations 1 Standard: 6.EE.1 No Calculator: Write and evaluate numerical expressions involving whole-number exponents The population of fruit flies increases by a factor of

### Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 173.

Algebra Chapter : Systems of Equations and Inequalities - Systems of Equations Chapter : Systems of Equations and Inequalities System of Linear Equations: - two or more linear equations on the same coordinate

### 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

### Math Placement and Quantitative Literacy Study Guide

Math Placement and Quantitative Literacy Study Guide If you have any questions concerning this document, please E-mail the Department of Mathematics. This review is not intended to cover all of the material

### To solve the given equation we proceed as follows: Given equation Remove parentheses. Subtract 4x

.1 Solving Linear Equations: Equation: is a statement that one algebraic expression is equal to another. Linear equation: is an equation that can be written in the form ax + b 0. Solve: means find all

### Grade 8 Linear Equations in One Variable 8.EE.7a-b

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 8 Linear Equations in One Variable 8.EE.7a-b 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATH

### Lesson Plan Mine Shaft Grade 8 Slope

CCSSM: Grade 8 DOMAIN: Expressions and Equations Cluster: Understand the connections between proportional relationships, lines, and linear equations. Standard: 8.EE.5: Graph proportional relationships,

### Topic 15 Solving System of Linear Equations

Topic 15 Solving System of Linear Equations Introduction: A Linear Equation is an equation that contains two variables, typically x and y that when they are graphed on a cartesian grid make a straight

### 2.1 Algebraic Expressions and Combining like Terms

2.1 Algebraic Expressions and Combining like Terms Evaluate the following algebraic expressions for the given values of the variables. 3 3 3 Simplify the following algebraic expressions by combining like

### Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

### Systems of Linear Equations. Suggested Time: 11 Hours

Systems of Linear Equations Suggested Time: 11 Hours Unit Overview Focus and Context In this unit, students will model a situation with a linear system and use the equations to solve a related problem.

### From the Webisode: Math Meets Fashion

lesson CCSS CONNECTIONS Percent Markups From the Webisode: Math Meets Fashion In this lesson, s solve a multi-step problem by identifying percent markups of a whole and calculating a final sale price.

### Modeling Situations with Linear Equations

Modeling Situations with Linear Equations MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students use algebra in context, and in particular, how well students: Explore relationships

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### REVIEW: Write each statement as an inequality and then graph the inequality.

LESSON 1-5 NOTES (Part A): SOLVING INEQUALITIES Words like "at most" and "at least" suggest a relationship in which two quantities may not be equal. These relationships can be represented by a mathematical

### 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

### Isolate the absolute value expression. Add 5 to both sides and then divide by 2.

11 of 21 8/14/2014 2:35 PM of a variable expression. You can use the definition of absolute value to solve absolute value equations algebraically. Since then the equation ax + b = c is equivalent to (ax

### We can use the last column to set up an equation involving total price in order to solve for x.

FOOD Tasha ordered soup and salad for lunch. The soup cost 15 cents per ounce, and the salad cost 20 cents per ounce. If Tasha ordered 10 ounces of soup for lunch and the total cost was \$3.30, how many

### Linear Equations in One Variable

Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this section we will learn how to: Recognize and combine like terms. Solve

### 5.2. Systems of linear equations and their solution sets

5.2. Systems of linear equations and their solution sets Solution sets of systems of equations as intersections of sets Any collection of two or more equations is called a system of equations. The solution

### Helpsheet. Giblin Eunson Library LINEAR EQUATIONS. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet Giblin Eunson Library LINEAR EQUATIONS Use this sheet to help you: Solve linear equations containing one unknown Recognize a linear function, and identify its slope and intercept parameters Recognize

### Unit 1 Equations, Inequalities, Functions

Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1-100 Overview: This unit models real-world situations by using one- and two-variable linear equations. This unit will further expand upon pervious

### How does the decimal point move when you rewrite a percent as a decimal and when you rewrite a decimal as a percent?

6.1 Percents and Decimals How does the decimal point move when you rewrite a percent as a decimal and when you rewrite a decimal as a percent? 1 ACTIVITY: Writing Percents as Decimals Work with a partner.

### Solution of the System of Linear Equations: any ordered pair in a system that makes all equations true.

Definitions: Sstem of Linear Equations: or more linear equations Sstem of Linear Inequalities: or more linear inequalities Solution of the Sstem of Linear Equations: an ordered pair in a sstem that makes

### 1 Solve problems using. 2 Use functions to model. 3 Perform a break-even SECTION PROBLEM SOLVING AND BUSINESS APPLICATIONS USING SYSTEMS OF EQUATIONS

178 Chapter 3 Systems of Linear Equations SECTION 3.2 PROBLEM SOLVING AND BUSINESS APPLICATIONS USING SYSTEMS OF EQUATIONS Objectives 1 Solve problems using systems of equations. 2 Use functions to model

### 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

### Topic 2 Solving Equations

Topic 2 Solving Equations Introduction: When you are given the value of a variable and an algebraic expression then you can evaluate the expression. For example, If you are told that x = 6 then the value

### Data Chapter Questions. 1. How do scatter plots help us analyze data from bivariate data?

Data Chapter Questions 1. How do scatter plots help us analyze data from bivariate data? 2. What is the prediction equation and what information can it tell us? 3. What is a two-way table? What is relative

1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

### DIAMOND PROBLEMS 1.1.1

DIAMOND PROBLEMS 1.1.1 In every Diamond Problem, the product of the two side numbers (left and right) is the top number and their sum is the bottom number. Diamond Problems are an excellent way of practicing

### 1 You Try. 2 You Try.

1 Simplify. 1 You Try. 1) Simplify. AF 1.2 2 Evaluate if, and. 2 You Try. 2) Evaluate if. 1.0 Page 1 of 19 MDC@ACOE (AUSD) 09/13/10 3 Simplify. 3 You Try. 3) Simplify. NS 1.2 4 Simplify. 4 You Try. 4)

### Word Problems Involving Systems of Linear Equations

Word Problems Involving Systems of Linear Equations Many word problems will give rise to systems of equations that is, a pair of equations like this: 2x+3y = 10 x 6y = 5 You can solve a system of equations

### 3.1. Solving Two-Step Equations. Using Subtraction and Division to Solve. Goal: Solve two-step equations.

3.1 Solving Two-Step Equations Goal: Solve two-step equations. Notice in Example 1 that you isolate x by working backward. First you subtract from each side and then you divide. Example 1 Using Subtraction

### 1 of 43. Simultaneous Equations

1 of 43 Simultaneous Equations Simultaneous Equations (Graphs) There is one pair of values that solves both these equations: x + y = 3 y x = 1 We can find the pair of values by drawing the lines x + y

### Solving Systems by Elimination 35

10/21/13 Solving Systems by Elimination 35 EXAMPLE: 5x + 2y = 1 x 3y = 7 1.Multiply the Top equation by the coefficient of the x on the bottom equation and write that equation next to the first equation

### CD 1 Real Numbers, Variables, and Algebraic Expressions

CD 1 Real Numbers, Variables, and Algebraic Expressions The Algebra I Interactive Series is designed to incorporate all modalities of learning into one easy to use learning tool; thereby reinforcing learning

### GASOLINE The graph represents the cost of gasoline at \$3 per gallon.

9-6 Slope-Intercept Form MAIN IDEA Graph linear equations using the slope and -intercept. New Vocabular slope-intercept form -intercept Math Online glencoe.com Etra Eamples Personal Tutor Self-Check Quiz

### Solving Two-Step and Multi-Step Equations

-3 Solving Two-Step and Multi-Step Equations Objective Solve equations in one variable that contain more than one operation. Why learn this? Equations containing more than one operation can model real-world

### Analytic Geometry Section 2-6: Circles

Analytic Geometry Section 2-6: Circles Objective: To find equations of circles and to find the coordinates of any points where circles and lines meet. Page 81 Definition of a Circle A circle is the set