Lesson 3-2: Solving Linear Systems Algebraically

Size: px
Start display at page:

Download "Lesson 3-2: Solving Linear Systems Algebraically"

Transcription

1 Yesterday we took our first look at solving a linear system. We learned that a linear system is two or more linear equations taken at the same time. Their solution is the point that all the lines have in common, their intersection. We also learned there are three ways to solve a linear system: 1. Graph the lines, visually locate the intersection: only approximates the solution. 2. Substitution: solve one equation for a variable, substitute into the other. 3. Combination: add the two equations together to eliminate one variable. Today we will work on the latter two; they will give us a precise answer, not an estimation. The basic idea If we have an equation with only one variable, we can solve that equation and find the value of the variable that solves the equation. The problem we have when trying to solve a linear system is that there s more than one variable. The basic idea when solving linear systems is to: 1. Eliminate one of the variables 2. Solve the resulting one variable equation 3. Plug the resulting value back into the original two variable equation to solve for the other variable. The trick is step one: eliminating one of the variables. That s what we ll be learning to day. Using substitution to solve a linear system The first way we can eliminate one of the variables is with substitution. We use substitution to solve algebra problems all the time. Anytime you have an equation and you know the value of x, you use substitution to find y: If y = 3x 9 and x = -2, find y. y = 3x 9 y = 3(-2) 9 y = -6 9 y = -15 Here we substituted -2 for x in order to find the corresponding value of y. We can use the same technique to solve a linear system. Consider the following: y = 2x + 8 3x + y = -2 Page 1 of 6

2 Here we a linear system. The first equation is in slope-intercept form and the second is in standard form. Can you see a way to use substitution to eliminate one of the variables? Stop for a second and think; the next sentence has a hint. Hint: is there a way you can get rid of y? With the first equation, we know that y is the same thing as 2x + 8. What if in the second equation we substituted 2x + 8 in for y? That makes sense doesn t it? If y equals (or is the same as) 2x + 8 won t everything be the same if we replace y with 2x + 8? Let s try it: y = 2x + 8 3x + y = -2 3x + (2x + 8) = -2 5x + 8 = -2 5x = -10 x = -2 y = 2x + 8 y = 2(-2) + 8 y = y = 4 here we know what y is in terms of x we ll substitute that in for y in this equation to solve for x here we substituted (2x + 8) for y subtract 8 from both sides to get x alone divide both sides by 5 to get x alone we now know x we ll now find out what y is when x = -2 substitute -2 in for x and we know have the y value too The solution for this linear system is (-2, 4). Is there a way we can double check this? Sure! Actually, there are two ways: 1. Graph the lines 2. Plug this x & y value pair into both equations to verify it works for both. We ll do number 2 here: Equation1 Equation 2 y 2x 8 3x y 2 4 2( 2) 8 3( 2) Both check out so we can be confident that (-2, 4) is the solution for this linear system. Try substitution again Find the solution for the following linear system; be careful, it s a little different! To get started, take a look at this linear system and compare it to the one we just did. 5x 3y = 2 x + 2y = 3 Page 2 of 6

3 What is different? Here both equations are in standard form? What enabled us to do substitution in the first example? It was the fact that one of the equations was already solved for one of the variables. In the prior example, the first equation was already solved for y. So what we need to do here is solve one of the equations for a variable. It really doesn t matter if we solve for x or y; pick the easiest one. Which of the two equations do you think will be easiest to solve for a variable, and which variable will you solve it for? I d pick the second equation and solve it for x. Why? Because that is the only equation with a variable all by itself (it has a coefficient of 1). That will make it super easy to solve for x: just subtract 2y from both sides and you re done! This gives us x = -2y + 3. What is our next step? Plug that expression into the first equation in place of x: 5x 3y = 2 start with the first equation 5(-2y + 3) 3y = 2 substitute (-2y + 3) in place of x -10y y = 2 use distributive property to get rid of parentheses -13y + 15 = 2-13y = -13 subtract 15 from both sides to get y by itself y = 1 divide both sides by -13 to get what y is equal to 5x 3y = 2 5x 3(1) = 2 5x - 3 = 2 5x = 5 x = 1 go back to the original 2 variable equation substitute the known value for y back into the equation add 3 to both sides to get x alone divide both sides by 5 to solve for x The solution for the linear system is (1, 1). Plug these values into both equations to check the answer. What are the steps for solving using substitution? 1. Solve one equation for one of its variables. 2. Substitute this expression into the other equation and solve for the other variable. 3. Substitute the (now) known value into either of the 2 variable equations & solve. 4. Check the solution in each of the original equations. Page 3 of 6

4 Using combination to solve linear systems The second method for solving linear systems is the combination method. In the combination method, we arrange the equations so that if we add them together one of the variables is cancelled out/eliminated. To do this we: 1. Arrange the two equations so like terms are aligned in columns. 2. Multiply one (or both) equations by a constant so that one of the variable sets differs only in sign. In other words, when you add them together they cancel out. 3. Add the equations and solve for the remaining variable. 4. Substitute the now known value of the solved variable back into either of the original equations and solve for the other variable. 5. Check your solution in each of the original equations. Let s try this out. Solve this linear system: 11x + 7y = 9 6x + 7y = 24 Step one: Arrange the two equations so like terms are aligned in columns. We already there the x s and the y s are aligned above each other. Step two: Multiply one (or both) equations by a constant so that one of the variable sest differs only in sign. We re close. In both equations y has a coefficient of 7. All we need to do is multiply one of the equations (both sides) by -1. I ll pick the first one: -1(11x + 7y) = -1(9) -11x 7y = -9 mult both sides by -1-11x 7y = -9 here s the adjusted 1 st equation 6x + 7y = 24 and here s the original 2 nd equation Notice how the y s have the same coefficient (7) but differ only in sign. Step three: Add the equations and solve for the remaining variable. To add the equations, just add like terms. This is easy since we ve arranged the equations so that like terms are aligned in columns. Page 4 of 6

5 11x 7 y 9 add thesetwo... 6x 7 y equations together 5x 0 y 15 this basically cancels the y out 5x 15 x 3 divideboth sides by 5to solve for x Step four: Substitute the now known value for the solved variable back into either of the original equations and solve for the other variable. I ll pick the second original equation because it has simpler coefficients: 6x 7 y 24 6( 3) 7 y y 24 7 y 42 y 6 Step five: Check your solution in each of the original equations: 11x 7y 9 6x 7y 24 11( 3) 7(6) 9 6( 3) 7(6) Both check out so the solution is (-3, 6). Parallel or same lines when using substitution or combination If when you substitute or combine: 1. The resulting equation is false (for example 2 = 3), the lines are parallel. 2. The resulting equation is true (for example 3 = 3), the lines are the same. Examples: 9x 12y 3 2x 4y 2 ( by 5) 3x 4y 2 ( by 3) 10x 20y 10 9x 12y 3 10x 20y 10 9x 12y 6 ( combine) 10x 20y 10 ( combine) 0 6 ( false; no solution) 0 0 ( true; number of solutions) Page 5 of 6

6 Cookbook for solving linear systems algebraically General guidelines 1. Use substitution if one of the equations has a variable with a coefficient of 1 or Use combination otherwise. Combination is easier if the equations already have a variable set with the same coefficient. 3. System has infinite number of solutions (lines are the same) if when you eliminate, both variables are canceled and you have a true statement. 4. System has no solution (lines are parallel) if when you eliminate, both variables are canceled and you have a false statement. Substitution cookbook 1. Solve one equation for one of its variables. 2. Substitute this expression into the other equation and solve for the other variable. 3. Substitute the (now) known value into either of the 2 variable equations & solve. 4. Check the solution in each of the original equations. Combination cookbook 1. Arrange the two equations so like terms are aligned in columns. 2. Multiply one (or both) equations by a constant so that one of the variable sets differs only in sign. In other words, when you add them together they cancel out. 3. Add the equations and solve for the remaining variable. 4. Substitute the now known value for the solved variable back into either of the original equations and solve for the other variable. 5. Check your solution in each of the original equations. Page 6 of 6

Lesson 3-1: Solving Linear Systems by Graphing

Lesson 3-1: Solving Linear Systems by Graphing For the past several weeks we ve been working with linear equations. We ve learned how to graph them and the three main forms they can take. Today we re going to begin considering what happens when we

More information

Definition: A "system" of equations is a set or collection of equations that you deal with all together at once.

Definition: A system of equations is a set or collection of equations that you deal with all together at once. System of Equations Definition: A "system" of equations is a set or collection of equations that you deal with all together at once. There is both an x and y value that needs to be solved for Systems

More information

SNAP Centre Workshop. Solving Systems of Equations

SNAP Centre Workshop. Solving Systems of Equations SNAP Centre Workshop Solving Systems of Equations 35 Introduction When presented with an equation containing one variable, finding a solution is usually done using basic algebraic manipulation. Example

More information

Chapter 6. Systems of Equations and Inequalities

Chapter 6. Systems of Equations and Inequalities Chapter 6 Systems of Equations and Inequalities 6.1 Solve Linear Systems by Graphing I can graph and solve systems of linear equations. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.6 What is a system

More information

Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables

Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables Solve Systems of Linear Equations by Graphing Solve Systems of Linear Equations by the Substitution Method Solve Systems of

More information

Chapter 1 Review of Equations and Inequalities

Chapter 1 Review of Equations and Inequalities Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve

More information

Distributive property and its connection to areas

Distributive property and its connection to areas February 27, 2009 Distributive property and its connection to areas page 1 Distributive property and its connection to areas Recap: distributive property The distributive property says that when you multiply

More information

Systems of Linear Equations and Inequalities

Systems of Linear Equations and Inequalities Systems of Linear Equations and Inequalities Alex Moore February 4, 017 1 What is a system? Now that we have studied linear equations and linear inequalities, it is time to consider the question, What

More information

Consistent and Dependent

Consistent and Dependent Graphing a System of Equations System of Equations: Consists of two equations. The solution to the system is an ordered pair that satisfies both equations. There are three methods to solving a system;

More information

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

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

More information

Answers to the problems will be posted on the school website, go to Academics tab, then select Mathematics and select Summer Packets.

Answers to the problems will be posted on the school website, go to Academics tab, then select Mathematics and select Summer Packets. Name Geometry SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Geometry. We will use these concepts on a regular basis throughout

More information

2x + 5 = x = x = 4

2x + 5 = x = x = 4 98 CHAPTER 3 Algebra Textbook Reference Section 5.1 3.3 LINEAR EQUATIONS AND INEQUALITIES Student CD Section.5 CLAST OBJECTIVES Solve linear equations and inequalities Solve a system of two linear equations

More information

Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:

Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the

More information

A. Incorrect! Replacing is not a method for solving systems of equations.

A. Incorrect! Replacing is not a method for solving systems of equations. ACT Math and Science - Problem Drill 20: Systems of Equations No. 1 of 10 1. What methods were presented to solve systems of equations? (A) Graphing, replacing, and substitution. (B) Solving, replacing,

More information

= (1 3 )= =4 3 +2=4. Now that we have it down to a simple two-step equation, we can solve like normal and get the following: =4 2

= (1 3 )= =4 3 +2=4. Now that we have it down to a simple two-step equation, we can solve like normal and get the following: =4 2 6.3 Solving Systems with Substitution While graphing is useful for an estimate, the main way that we can solve a system to get an exact answer is algebraically. There are a few useful methods to do this,

More information

Along the way, you learned many manipulative skills using the Properties of Real Numbers.

Along the way, you learned many manipulative skills using the Properties of Real Numbers. A LOOK at Algebra ============================= In a nutshell, you have learned how to: 1. solve linear equations and inequalities. solve quadratic equations and inequalities 3. solve systems of linear

More information

Conceptual Explanations: Simultaneous Equations Distance, rate, and time

Conceptual Explanations: Simultaneous Equations Distance, rate, and time Conceptual Explanations: Simultaneous Equations Distance, rate, and time If you travel 30 miles per hour for 4 hours, how far do you go? A little common sense will tell you that the answer is 120 miles.

More information

Solving Systems of Equations

Solving Systems of Equations Solving Systems of Equations Solving Systems of Equations What are systems of equations? Two or more equations that have the same variable(s) Solving Systems of Equations There are three ways to solve

More information

Name Period Date Ch. 5 Systems of Linear Equations Review Guide

Name Period Date Ch. 5 Systems of Linear Equations Review Guide Reteaching 5-1 Solving Systems by Graphing ** A system of equations is a set of two or more equations that have the same variables. ** The solution of a system is an ordered pair that satisfies all equations

More information

ACCUPLACER MATH 0311 OR MATH 0120

ACCUPLACER MATH 0311 OR MATH 0120 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises

More information

Lesson 2-6: Graphs of Absolute Value Equations

Lesson 2-6: Graphs of Absolute Value Equations Where we re headed today Today we re going to take the net graphing step we ll learn how to graph absolute value equations. Here are the three things you are going to need to be able to do: 1. Match an

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations Section 2.3 Solving Systems of Linear Equations TERMINOLOGY 2.3 Previously Used: Equivalent Equations Literal Equation Properties of Equations Substitution Principle Prerequisite Terms: Coordinate Axes

More information

O.K. But what if the chicken didn t have access to a teleporter.

O.K. But what if the chicken didn t have access to a teleporter. The intermediate value theorem, and performing algebra on its. This is a dual topic lecture. : The Intermediate value theorem First we should remember what it means to be a continuous function: A function

More information

Section 4.6 Negative Exponents

Section 4.6 Negative Exponents Section 4.6 Negative Exponents INTRODUCTION In order to understand negative exponents the main topic of this section we need to make sure we understand the meaning of the reciprocal of a number. Reciprocals

More information

Chapter 2 Linear Equations and Inequalities in One Variable

Chapter 2 Linear Equations and Inequalities in One Variable Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound

More information

Algebra 2 Honors Unit 1 Review of Algebra 1

Algebra 2 Honors Unit 1 Review of Algebra 1 Algebra Honors Unit Review of Algebra Day Combining Like Terms and Distributive Property Objectives: SWBAT evaluate and simplify expressions involving real numbers. SWBAT evaluate exponents SWBAT combine

More information

Ch. 3 Equations and Inequalities

Ch. 3 Equations and Inequalities Ch. 3 Equations and Inequalities 3.1 Solving Linear Equations Graphically There are 2 methods presented in this section for solving linear equations graphically. Normally I would not cover solving linear

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Lines and Their Equations

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Lines and Their Equations ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 017/018 DR. ANTHONY BROWN. Lines and Their Equations.1. Slope of a Line and its y-intercept. In Euclidean geometry (where

More information

Pre-calculus is the stepping stone for Calculus. It s the final hurdle after all those years of

Pre-calculus is the stepping stone for Calculus. It s the final hurdle after all those years of Chapter 1 Beginning at the Very Beginning: Pre-Pre-Calculus In This Chapter Brushing up on order of operations Solving equalities Graphing equalities and inequalities Finding distance, midpoint, and slope

More information

3: Linear Systems. Examples. [1.] Solve. The first equation is in blue; the second is in red. Here's the graph: The solution is ( 0.8,3.4 ).

3: Linear Systems. Examples. [1.] Solve. The first equation is in blue; the second is in red. Here's the graph: The solution is ( 0.8,3.4 ). 3: Linear Systems 3-1: Graphing Systems of Equations So far, you've dealt with a single equation at a time or, in the case of absolute value, one after the other. Now it's time to move to multiple equations

More information

Algebra Review. Finding Zeros (Roots) of Quadratics, Cubics, and Quartics. Kasten, Algebra 2. Algebra Review

Algebra Review. Finding Zeros (Roots) of Quadratics, Cubics, and Quartics. Kasten, Algebra 2. Algebra Review Kasten, Algebra 2 Finding Zeros (Roots) of Quadratics, Cubics, and Quartics A zero of a polynomial equation is the value of the independent variable (typically x) that, when plugged-in to the equation,

More information

Graphing Linear Systems

Graphing Linear Systems Graphing Linear Systems Goal Estimate the solution of a system of linear equations by graphing. VOCABULARY System of linear equations A system of linear equations is two or more linear equations in the

More information

Lesson 21 Not So Dramatic Quadratics

Lesson 21 Not So Dramatic Quadratics STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has

More information

Objective. The student will be able to: solve systems of equations using elimination with multiplication. SOL: A.9

Objective. The student will be able to: solve systems of equations using elimination with multiplication. SOL: A.9 Objective The student will be able to: solve systems of equations using elimination with multiplication. SOL: A.9 Designed by Skip Tyler, Varina High School Solving Systems of Equations So far, we have

More information

Summer Review. For Students Entering. Algebra 2 & Analysis

Summer Review. For Students Entering. Algebra 2 & Analysis Lawrence High School Math Department Summer Review For Students Entering Algebra 2 & Analysis Fraction Rules: Operation Explanation Example Multiply Fractions Multiply both numerators and denominators

More information

Solving and Graphing Inequalities

Solving and Graphing Inequalities Solving and Graphing Inequalities Graphing Simple Inequalities: x > 3 When finding the solution for an equation we get one answer for x. (There is only one number that satisfies the equation.) For 3x 5

More information

Reteach Simplifying Algebraic Expressions

Reteach Simplifying Algebraic Expressions 1-4 Simplifying Algebraic Expressions To evaluate an algebraic expression you substitute numbers for variables. Then follow the order of operations. Here is a sentence that can help you remember the order

More information

Math 121 (Lesieutre); 9.1: Polar coordinates; November 22, 2017

Math 121 (Lesieutre); 9.1: Polar coordinates; November 22, 2017 Math 2 Lesieutre; 9: Polar coordinates; November 22, 207 Plot the point 2, 2 in the plane If you were trying to describe this point to a friend, how could you do it? One option would be coordinates, but

More information

9.4 Radical Expressions

9.4 Radical Expressions Section 9.4 Radical Expressions 95 9.4 Radical Expressions In the previous two sections, we learned how to multiply and divide square roots. Specifically, we are now armed with the following two properties.

More information

Section 2.7 Solving Linear Inequalities

Section 2.7 Solving Linear Inequalities Section.7 Solving Linear Inequalities Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Add and multiply an inequality. Solving equations (.1,.,

More information

Sec. 1 Simplifying Rational Expressions: +

Sec. 1 Simplifying Rational Expressions: + Chapter 9 Rational Epressions Sec. Simplifying Rational Epressions: + The procedure used to add and subtract rational epressions in algebra is the same used in adding and subtracting fractions in 5 th

More information

System of Linear Equation: with more than Two Equations and more than Two Unknowns

System of Linear Equation: with more than Two Equations and more than Two Unknowns System of Linear Equation: with more than Two Equations and more than Two Unknowns Michigan Department of Education Standards for High School: Standard 1: Solve linear equations and inequalities including

More information

Introduction. So, why did I even bother to write this?

Introduction. So, why did I even bother to write this? Introduction This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The review contains the occasional

More information

MATCHING. Match the correct vocabulary word with its definition

MATCHING. Match the correct vocabulary word with its definition Name Algebra I Block UNIT 2 STUDY GUIDE Ms. Metzger MATCHING. Match the correct vocabulary word with its definition 1. Whole Numbers 2. Integers A. A value for a variable that makes an equation true B.

More information

Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities

Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,

More information

REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y?

REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y? REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate

More information

Algebra & Trig Review

Algebra & Trig Review Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The

More information

Math 138: Introduction to solving systems of equations with matrices. The Concept of Balance for Systems of Equations

Math 138: Introduction to solving systems of equations with matrices. The Concept of Balance for Systems of Equations Math 138: Introduction to solving systems of equations with matrices. Pedagogy focus: Concept of equation balance, integer arithmetic, quadratic equations. The Concept of Balance for Systems of Equations

More information

STEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is

STEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is EQUATIONS OF LINES 1. Writing Equations of Lines There are many ways to define a line, but for today, let s think of a LINE as a collection of points such that the slope between any two of those points

More information

Geometry 21 Summer Work Packet Review and Study Guide

Geometry 21 Summer Work Packet Review and Study Guide Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the

More information

Alex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1

Alex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1 Alex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1 What is a linear equation? It sounds fancy, but linear equation means the same thing as a line. In other words, it s an equation

More information

Lesson 28: Another Computational Method of Solving a Linear System

Lesson 28: Another Computational Method of Solving a Linear System Lesson 28: Another Computational Method of Solving a Linear System Student Outcomes Students learn the elimination method for solving a system of linear equations. Students use properties of rational numbers

More information

Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables

Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Department of Mathematics Porterville College September 7, 2014 Systems of Linear Equations in Two Variables Learning Objectives: Solve Systems

More information

Contents. To the Teacher... v

Contents. To the Teacher... v Katherine & Scott Robillard Contents To the Teacher........................................... v Linear Equations................................................ 1 Linear Inequalities..............................................

More information

We are working with degree two or

We are working with degree two or page 4 4 We are working with degree two or quadratic epressions (a + b + c) and equations (a + b + c = 0). We see techniques such as multiplying and factoring epressions and solving equations using factoring

More information

Week 7 Algebra 1 Assignment:

Week 7 Algebra 1 Assignment: Week 7 Algebra 1 Assignment: Day 1: Chapter 3 test Day 2: pp. 132-133 #1-41 odd Day 3: pp. 138-139 #2-20 even, 22-26 Day 4: pp. 141-142 #1-21 odd, 25-30 Day 5: pp. 145-147 #1-25 odd, 33-37 Notes on Assignment:

More information

Part 2 - Beginning Algebra Summary

Part 2 - Beginning Algebra Summary Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian

More information

MA094 Part 2 - Beginning Algebra Summary

MA094 Part 2 - Beginning Algebra Summary MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page

More information

Unit 5 Solving Quadratic Equations

Unit 5 Solving Quadratic Equations SM Name: Period: Unit 5 Solving Quadratic Equations 5.1 Solving Quadratic Equations by Factoring Quadratic Equation: Any equation that can be written in the form a b c + + = 0, where a 0. Zero Product

More information

Solving Linear Equations - One Step Equations

Solving Linear Equations - One Step Equations 1.1 Solving Linear Equations - One Step Equations Objective: Solve one step linear equations by balancing using inverse operations Solving linear equations is an important and fundamental skill in algebra.

More information

Name Period Date. ** A system of equations is a set of two or more equations that have the same variables.

Name Period Date. ** A system of equations is a set of two or more equations that have the same variables. Reteaching 5-1 Solving Systems by Graphing ** A system of equations is a set of two or more equations that have the same variables. ** The solution of a system is an ordered pair that satisfies all equations

More information

Contents. To the Teacher... v

Contents. To the Teacher... v Katherine & Scott Robillard Contents To the Teacher........................................... v Linear Equations................................................ 1 Linear Inequalities..............................................

More information

UNIT 3 REASONING WITH EQUATIONS Lesson 2: Solving Systems of Equations Instruction

UNIT 3 REASONING WITH EQUATIONS Lesson 2: Solving Systems of Equations Instruction Prerequisite Skills This lesson requires the use of the following skills: graphing equations of lines using properties of equality to solve equations Introduction Two equations that are solved together

More information

MA 1128: Lecture 08 03/02/2018. Linear Equations from Graphs And Linear Inequalities

MA 1128: Lecture 08 03/02/2018. Linear Equations from Graphs And Linear Inequalities MA 1128: Lecture 08 03/02/2018 Linear Equations from Graphs And Linear Inequalities Linear Equations from Graphs Given a line, we would like to be able to come up with an equation for it. I ll go over

More information

2.4 Graphing Inequalities

2.4 Graphing Inequalities .4 Graphing Inequalities Why We Need This Our applications will have associated limiting values - and either we will have to be at least as big as the value or no larger than the value. Why We Need This

More information

base 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.

base 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation. EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences

More information

Algebra Exam. Solutions and Grading Guide

Algebra Exam. Solutions and Grading Guide Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full

More information

Cool Results on Primes

Cool Results on Primes Cool Results on Primes LA Math Circle (Advanced) January 24, 2016 Recall that last week we learned an algorithm that seemed to magically spit out greatest common divisors, but we weren t quite sure why

More information

Solve Systems of Equations Algebraically

Solve Systems of Equations Algebraically Part 1: Introduction Solve Systems of Equations Algebraically Develop Skills and Strategies CCSS 8.EE.C.8b You know that solutions to systems of linear equations can be shown in graphs. Now you will learn

More information

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation? Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using

More information

Chapter 3 ALGEBRA. Overview. Algebra. 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents. Section 3.

Chapter 3 ALGEBRA. Overview. Algebra. 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents. Section 3. 4 Chapter 3 ALGEBRA Overview Algebra 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents 5 LinearEquations 3+ what = 7? If you have come through arithmetic, the

More information

Honors Advanced Mathematics Determinants page 1

Honors Advanced Mathematics Determinants page 1 Determinants page 1 Determinants For every square matrix A, there is a number called the determinant of the matrix, denoted as det(a) or A. Sometimes the bars are written just around the numbers of the

More information

Math Lecture 18 Notes

Math Lecture 18 Notes Math 1010 - Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,

More information

irst we need to know that there are many ways to indicate multiplication; for example the product of 5 and 7 can be written in a variety of ways:

irst we need to know that there are many ways to indicate multiplication; for example the product of 5 and 7 can be written in a variety of ways: CH 2 VARIABLES INTRODUCTION F irst we need to know that there are many ways to indicate multiplication; for example the product of 5 and 7 can be written in a variety of ways: 5 7 5 7 5(7) (5)7 (5)(7)

More information

Math 154 :: Elementary Algebra

Math 154 :: Elementary Algebra Math 4 :: Elementary Algebra Section. Additive Property of Equality Section. Multiplicative Property of Equality Section.3 Linear Equations in One-Variable Section.4 Linear Equations in One-Variable with

More information

A Quick Algebra Review

A Quick Algebra Review 1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

More information

Quadratic Equations Part I

Quadratic Equations Part I Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing

More information

Exponents. Reteach. Write each expression in exponential form (0.4)

Exponents. Reteach. Write each expression in exponential form (0.4) 9-1 Exponents You can write a number in exponential form to show repeated multiplication. A number written in exponential form has a base and an exponent. The exponent tells you how many times a number,

More information

Chapter 2A - Solving Equations

Chapter 2A - Solving Equations - Chapter A Chapter A - Solving Equations Introduction and Review of Linear Equations An equation is a statement which relates two or more numbers or algebraic expressions. For example, the equation 6

More information

Unit 6 Study Guide: Equations. Section 6-1: One-Step Equations with Adding & Subtracting

Unit 6 Study Guide: Equations. Section 6-1: One-Step Equations with Adding & Subtracting Unit 6 Study Guide: Equations DUE DATE: A Day: Dec 18 th B Day: Dec 19 th Name Period Score / Section 6-1: One-Step Equations with Adding & Subtracting Textbook Reference: Page 437 Vocabulary: Equation

More information

ACCUPLACER MATH 0310

ACCUPLACER MATH 0310 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

More information

Q520: Answers to the Homework on Hopfield Networks. 1. For each of the following, answer true or false with an explanation:

Q520: Answers to the Homework on Hopfield Networks. 1. For each of the following, answer true or false with an explanation: Q50: Answers to the Homework on Hopfield Networks 1. For each of the following, answer true or false with an explanation: a. Fix a Hopfield net. If o and o are neighboring observation patterns then Φ(

More information

Module 2 Study Guide. The second module covers the following sections of the textbook: , 4.1, 4.2, 4.5, and

Module 2 Study Guide. The second module covers the following sections of the textbook: , 4.1, 4.2, 4.5, and Module 2 Study Guide The second module covers the following sections of the textbook: 3.3-3.7, 4.1, 4.2, 4.5, and 5.1-5.3 Sections 3.3-3.6 This is a continuation of the study of linear functions that we

More information

Rational Expressions & Equations

Rational Expressions & Equations Chapter 9 Rational Epressions & Equations Sec. 1 Simplifying Rational Epressions We simply rational epressions the same way we simplified fractions. When we first began to simplify fractions, we factored

More information

How much does the cow weigh?

How much does the cow weigh? 1 GRADE 11 PRE-CALCULUS UNIT G SYSTEMS UNIT NOTES 1. Solving Systems of equation is an important subject. To date you have only learned to solve for one unknown in an equation for example: when does 2x

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

Chapter One: Pre-Geometry

Chapter One: Pre-Geometry Chapter One: Pre-Geometry Index: A: Solving Equations B: Factoring (GCF/DOTS) C: Factoring (Case Two leading into Case One) D: Factoring (Case One) E: Solving Quadratics F: Parallel and Perpendicular Lines

More information

Algebra. Robert Taggart

Algebra. Robert Taggart Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit 1: Algebra Basics Lesson 1: Negative and Positive Numbers....................... Lesson 2: Operations

More information

The trick is to multiply the numerator and denominator of the big fraction by the least common denominator of every little fraction.

The trick is to multiply the numerator and denominator of the big fraction by the least common denominator of every little fraction. Complex Fractions A complex fraction is an expression that features fractions within fractions. To simplify complex fractions, we only need to master one very simple method. Simplify 7 6 +3 8 4 3 4 The

More information

Matrices, Row Reduction of Matrices

Matrices, Row Reduction of Matrices Matrices, Row Reduction of Matrices October 9, 014 1 Row Reduction and Echelon Forms In the previous section, we saw a procedure for solving systems of equations It is simple in that it consists of only

More information

Lesson 3: Using Linear Combinations to Solve a System of Equations

Lesson 3: Using Linear Combinations to Solve a System of Equations Lesson 3: Using Linear Combinations to Solve a System of Equations Steps for Using Linear Combinations to Solve a System of Equations 1. 2. 3. 4. 5. Example 1 Solve the following system using the linear

More information

5.2 Infinite Series Brian E. Veitch

5.2 Infinite Series Brian E. Veitch 5. Infinite Series Since many quantities show up that cannot be computed exactly, we need some way of representing it (or approximating it). One way is to sum an infinite series. Recall that a n is the

More information

Graphical Solutions of Linear Systems

Graphical Solutions of Linear Systems Graphical Solutions of Linear Systems Consistent System (At least one solution) Inconsistent System (No Solution) Independent (One solution) Dependent (Infinite many solutions) Parallel Lines Equations

More information

7.1 Solving Linear Systems by Graphing

7.1 Solving Linear Systems by Graphing 7.1 Solving Linear Sstems b Graphing Objectives: Learn how to solve a sstem of linear equations b graphing Learn how to model a real-life situation using a sstem of linear equations With an equation, an

More information

Chapter 5 Simplifying Formulas and Solving Equations

Chapter 5 Simplifying Formulas and Solving Equations Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L + W + L + W. Can this formula be written in a simpler way? If it is true, that we can

More information

CP Algebra 2. Unit 3B: Polynomials. Name: Period:

CP Algebra 2. Unit 3B: Polynomials. Name: Period: CP Algebra 2 Unit 3B: Polynomials Name: Period: Learning Targets 10. I can use the fundamental theorem of algebra to find the expected number of roots. Solving Polynomials 11. I can solve polynomials by

More information

How can we see it? Is there another way?

How can we see it? Is there another way? You have solved a variety of equations in this course. In this lesson, by looking at equations in different ways, you will be able to solve even more complicated equations quickly and easily. These new

More information

Intersecting Two Lines, Part Two

Intersecting Two Lines, Part Two Module 1.5 Page 149 of 1390. Module 1.5: Intersecting Two Lines, Part Two In this module you will learn about two very common algebraic methods for intersecting two lines: the Substitution Method and the

More information

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY 2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you

More information

Math 2 Variable Manipulation Part 6 System of Equations

Math 2 Variable Manipulation Part 6 System of Equations Name: Date: 1 Math 2 Variable Manipulation Part 6 System of Equations SYSTEM OF EQUATIONS INTRODUCTION A "system" of equations is a set or collection of equations that you deal with all together at once.

More information