# 10.1 Systems of Linear Equations: Substitution and Elimination

Save this PDF as:

Size: px
Start display at page:

Download "10.1 Systems of Linear Equations: Substitution and Elimination"

## Transcription

1 726 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 (Appendix, Section A.5, pp ) Lines (Section 1.4, pp 27 38) Now work the Are You Prepared? prolems on page 737. OBJECTIVES 1 Solve Systems of Equations y Sustitution 2 Solve Systems of Equations y Elimination 3 Identify Inconsistent Systems of Equations Containing Two Variales 4 Express the of a System of Dependent Equations Containing Two Variales 5 Solve Systems of Three Equations Containing Three Variales 6 Identify Inconsistent Systems of Equations Containing Three Variales 7 Express the of a System of Dependent Equations Containing Three Variales We egin with an example. EXAMPLE 1 Movie Theater Ticket Sales A movie theater sells tickets for \$8.00 each, with seniors receiving a discount of \$2.00. One evening the theater took in \$3580 in revenue. If x represents the numer of tickets sold at \$8.00 and y the numer of tickets sold at the discounted price of \$6.00, write an equation that relates these variales. Each nondiscounted ticket rings in \$8.00, so x tickets will ring in 8x dollars. Similarly, y discounted tickets ring in 6y dollars. Since the total rought in is \$3580, we must have 8x + 6y = 3580 In Example 1, suppose that we also know that 525 tickets were sold that evening. Then we have another equation relating the variales x and y. The two equations x + y = 525 8x + 6y = 3580 x + y = 525 form a system of equations. In general, a system of equations is a collection of two or more equations, each containing one or more variales. Example 2 gives some illustrations of systems of equations.

2 SECTION 10.1 Systems of Linear Equations: Sustitution and Elimination 727 EXAMPLE 2 Examples of Systems of Equations (a) 2x + y = 5-4x + 6y = -2 Two equations containing two variales, x and y () x + y2 = 5 2x + y = 4 Two equations containing two variales, x and y (c) x + y + z = 6 c 3x - 2y + 4z = 9 x - y - z = 0 Three equations containing three variales, x, y, and z (d) x + y + z = 5 x - y = 2 Two equations containing three variales, x, y, and z (e) x + y + z = 6 2x + 2z = 4 d y + z = 2 x = 4 Four equations containing three variales, x, y, and z (4) We use a race, as shown, to remind us that we are dealing with a system of equations. We also will find it convenient to numer each equation in the system. A solution of a system of equations consists of values for the variales that are solutions of each equation of the system. To solve a system of equations means to find all solutions of the system. For example, x = 2, y = 1 is a solution of the system in Example 2(a), ecause 2x + y = 5-4x + 6y = = = = = -2 A solution of the system in Example 2() is x = 1, y = 2, ecause x + y2 = 5 2x + y = = = = = 4 Another solution of the system in Example 2() is x = 11 which you can 4, y = - 3 2, check for yourself. A solution of the system in Example 2(c) is x = 3, y = 2, z = 1, ecause x + y + z = 6 c 3x - 2y + 4z = 9 x - y - z = = 6 c = = = 0 Note that x = 3, y = 3, z = 0 is not a solution of the system in Example 2(c). x + y + z = 6 c 3x - 2y + 4z = 9 x - y - z = = 6 c = 3 Z = 0

3 728 CHAPTER 10 Systems of Equations and Inequalities Although these values satisfy equations and, they do not satisfy equation. Any solution of the system must satisfy each equation of the system. NOW WORK PROBLEM 9. When a system of equations has at least one solution, it is said to e consistent; otherwise, it is called inconsistent. An equation in n variales is said to e linear if it is equivalent to an equation of the form a 1 x 1 + a 2 x 2 + Á + a n x n = where x 1, x 2, Á, x n are n distinct variales, a 1, a 2, Á, a n, are constants, and at least one of the a s is not 0. Some examples of linear equations are 2x + 3y = 2 5x - 2y + 3z = 10 8x + 8y - 2z + 5w = 0 If each equation in a system of equations is linear, then we have a system of linear equations. The systems in Examples 2(a), (c), (d), and (e) are linear, whereas the system in Example 2() is nonlinear. In this chapter we shall solve linear systems in Sections 10.1 to We discuss nonlinear systems in Section We egin y discussing a system of two linear equations containing two variales.we can view the prolem of solving such a system as a geometry prolem.the graph of each equation in such a system is a line. So a system of two equations containing two variales represents a pair of lines. The lines either intersect or are parallel or are coincident (that is, identical). 1. If the lines intersect, then the system of equations has one solution, given y the point of intersection. The system is consistent and the equations are independent. See Figure 1(a). 2. If the lines are parallel, then the system of equations has no solution, ecause the lines never intersect. The system is inconsistent. See Figure 1(). 3. If the lines are coincident, then the system of equations has infinitely many solutions, represented y the totality of points on the line. The system is consistent and the equations are dependent. See Figure 1(c). Figure 1 y y y x x x (a) Intersecting lines; system has one solution () Parallel lines; system has no solution (c) Coincident lines; system has infinitely many solutions EXAMPLE 3 Solving a System of Linear Equations Using a Graphing Utility 2x + y = 5-4x + 6y = 12

4 SECTION 10.1 Systems of Linear Equations: Sustitution and Elimination 729 Figure 2 Y 1 2x Y 2 3x 2 1 First, we solve each equation for y. This is equivalent to writing each equation in slope intercept form. Equation in slope intercept form is Y 1 = -2x + 5. Equation in slope intercept form is Y 2 = 2 Figure 2 shows the graphs 3 x + 2. using a graphing utility. From the graph in Figure 2, we see that the lines intersect, so the system is consistent and the equations are independent. Using INTERSECT, we otain the solution , Solve Systems of Equations y Sustitution Sometimes to otain exact solutions, we must use algeraic methods. A numer of methods are availale to us for solving systems of linear equations algeraically. In this section, we introduce two methods: sustitution and elimination. We illustrate the method of sustitution y solving the system given in Example 3. EXAMPLE 4 Solving a System of Linear Equations y Sustitution 2x + y = 5-4x + 6y = 12 We solve the first equation for y, otaining 2x + y = 5 y = -2x + 5 Sutract 2x from each side of. We sustitute this result for y into the second equation. The result is an equation containing just the variale x, which we can then solve. -4x + 6y = 12-4x x + 52 = 12-4x - 12x + 30 = 12-16x = -18 x = = 9 8 Sustitute y = -2x + 5 in. Remove parentheses. Comine like terms and sutract 30 from oth sides. Divide each side y -16. Once we know that x = 9 8, we can easily find the value of y y ack- 9 sustitution, that is, y sustituting use the first equation. 8 for x in one of the original equations. We will 2x + y = 5 y = -2x + 5 y = -2a Sutract 2x from each side. Sustitute x = 9 in. 8 = = 11 4 The solution of the system is x = 9 8 = 1.125, y = 11 4 = 2.75.

5 730 CHAPTER 10 Systems of Equations and Inequalities CHECK: d 2x + y = 5: 2a = = 20 4 = 5-4x + 6y = 12: -4a a 11 4 = = 24 2 = 12 The method used to solve the system in Example 4 is called sustitution. The steps to e used are outlined next. Steps for Solving y Sustitution STEP 1: Pick one of the equations and solve for one of the variales in terms of the remaining variales. STEP 2: Sustitute the result into the remaining equations. STEP 3: If one equation in one variale results, solve this equation. Otherwise repeat Steps 1 and 2 until a single equation with one variale remains. STEP 4: Find the values of the remaining variales y ack-sustitution. STEP 5: Check the solution found. NOW USE SUBSTITUTION TO WORK PROBLEM Solve Systems of Equations y Elimination A second method for solving a system of linear equations is the method of elimination. This method is usually preferred over sustitution if sustitution leads to fractions or if the system contains more than two variales. Elimination also provides the necessary motivation for solving systems using matrices (the suject of Section 10.2). The idea ehind the method of elimination is to replace the original system of equations y an equivalent system so that adding two of the equations eliminates a variale. The rules for otaining equivalent equations are the same as those studied earlier. However, we may also interchange any two equations of the system and/or replace any equation in the system y the sum (or difference) of that equation and a nonzero multiple of any other equation in the system. In Words When using elimination, we want to get the coefficients of one of the variales to e negatives of one another. Rules for Otaining an Equivalent System of Equations 1. Interchange any two equations of the system. 2. Multiply (or divide) each side of an equation y the same nonzero constant. 3. Replace any equation in the system y the sum (or difference) of that equation and a nonzero multiple of any other equation in the system. An example will give you the idea. As you work through the example, pay particular attention to the pattern eing followed. EXAMPLE 5 Solving a System of Linear Equations y Elimination 2x + 3y = 1 -x + y = -3

6 SECTION 10.1 Systems of Linear Equations: Sustitution and Elimination 731 We multiply each side of equation y 2 so that the coefficients of x in the two equations are negatives of one another. The result is the equivalent system 2x + 3y = 1-2x + 2y = -6 If we add equations and we otain an equation containing just the variale y, which we can solve. 2x + 3y = 1-2x + 2y = -6 5y = -5 y = -1 We ack-sustitute this value for y in equation and simplify. 2x + 3y = 1 2x = 1 2x = 4 x = 2 Sustitute y = -1 in. Simplify. Solve for x. The solution of the original system is x = 2, y = -1. We leave it to you to check the solution. The procedure used in Example 5 is called the method of elimination. Notice the pattern of the solution. First, we eliminated the variale x from the second equation. Then we ack-sustituted; that is, we sustituted the value found for y ack into the first equation to find x. NOW USE ELIMINATION TO WORK PROBLEM 19. Let s return to the movie theater example Example 1. Add and. Solve for y. EXAMPLE 6 Movie Theater Ticket Sales A movie theater sells tickets for \$8.00 each, with seniors receiving a discount of \$2.00. One evening the theater sold 525 tickets and took in \$3580 in revenue. How many of each type of ticket were sold? If x represents the numer of tickets sold at \$8.00 and y the numer of tickets sold at the discounted price of \$6.00, then the given information results in the system of equations 8x + 6y = 3580 x + y = 525 We use the method of elimination. First, multiply the second equation y -6, and then add the equations. 8x + 6y = x - 6y = x = 430 x = 215 Add the equations. Since x + y = 525, then y = x = = 310. So 215 nondiscounted tickets and 310 senior discount tickets were sold.

7 732 CHAPTER 10 Systems of Equations and Inequalities 3 Identify Inconsistent Systems of Equations Containing Two Variales The previous examples dealt with consistent systems of equations that had a single solution. The next two examples deal with two other possiilities that may occur, the first eing a system that has no solution. EXAMPLE 7 An Inconsistent System of Linear Equations 2x + y = 5 4x + 2y = 8 We choose to use the method of sustitution and solve equation for y. 2x + y = 5 y = -2x + 5 Sutract 2x from each side. Now sustitute y = -2x + 5 for y in equation and solve for x. 4x + 2y = 8 Figure 3 10 Y 1 2x 5 4x x + 52 = 8 4x - 4x + 10 = 8 0 # x = -2 Sustitute y = -2x + 5 in. Remove parentheses. Sutract 10 from oth sides. This equation has no solution. We conclude that the system itself has no solution and is therefore inconsistent Y 2 2x 4 Figure 3 illustrates the pair of lines whose equations form the system in Example 7. Notice that the graphs of the two equations are lines, each with slope -2; one has a y-intercept of 5, the other a y-intercept of 4. The lines are parallel and have no point of intersection. This geometric statement is equivalent to the algeraic statement that the system has no solution. 4 Express the of a System of Dependent Equations Containing Two Variales EXAMPLE 8 Solving a System of Dependent Equations 2x + y = 4-6x - 3y = -12 We choose to use the method of elimination. 2x + y = 4-6x - 3y = -12 6x + 3y = 12-6x - 3y = -12 6x + 3y = 12 0 = 0 Multiply each side of equation y 3. Replace equation y the sum of equations and.

8 SECTION 10.1 Systems of Linear Equations: Sustitution and Elimination 733 The original system is equivalent to a system containing one equation, so the equations are dependent. This means that any values of x and y for which 6x + 3y = 12 or, equivalently, 2x + y = 4 are solutions. For example, x = 2, y = 0; x = 0, y = 4; x = -2, y = 8; x = 4, y = -4; and so on, are solutions. There are, in fact, infinitely many values of x and y for which 2x + y = 4, so the original system has infinitely many solutions. We will write the solution of the original system either as where x can e any real numer, or as where y can e any real numer. y = -2x + 4 x = y + 2 Figure 4 Y 1 Y 2 2x Figure 4 illustrates the situation presented in Example 8. Notice that the graphs of the two equations are lines, each with slope -2 and each with y-intercept 4. The lines are coincident. Notice also that equation in the original system is -3 times equation, indicating that the two equations are dependent. For the system in Example 8, we can write down some of the infinite numer of solutions y assigning values to x and then finding y = -2x + 4. If x = -2, then y = = 8. If x = 0, then y = 4. If x = 2, then y = 0. The ordered pairs 1-2, 82, 10, 42, and 12, 02 are three of the points on the line in Figure 4. NOW WORK PROBLEMS 25 AND Solve Systems of Three Equations Containing Three Variales Just as with a system of two linear equations containing two variales, a system of three linear equations containing three variales also has either exactly one solution (a consistent system with independent equations), or no solution (an inconsistent system), or infinitely many solutions (a consistent system with dependent equations). We can view the prolem of solving a system of three linear equations containing three variales as a geometry prolem.the graph of each equation in such a system is a plane in space. A system of three linear equations containing three variales represents three planes in space. Figure 5 illustrates some of the possiilities. Figure 5 (a) Consistent system; one solution s () Consistent system; infinite numer of solutions (c) Inconsistent system; no solution

9 734 CHAPTER 10 Systems of Equations and Inequalities Recall that a solution to a system of equations consists of values for the variales that are solutions of each equation of the system. For example, x = 3, y = -1, z = -5 is a solution to the system of equations c x + y + z = -3 2x - 3y + 6z = -21-3x + 5y = = = = = -9-5 = -14 ecause these values of the variales are solutions of each equation. Typically, when solving a system of three linear equations containing three variales, we use the method of elimination. Recall that the idea ehind the method of elimination is to form equivalent equations so that adding two of the equations eliminates a variale. Let s see how elimination works on a system of three equations containing three variales. EXAMPLE 9 Solving a System of Three Linear Equations with Three Variales Use the method of elimination to solve the system of equations. x + y - z = -1 c 4x - 3y + 2z = 16 2x - 2y - 3z = 5 For a system of three equations, we attempt to eliminate one variale at a time, using pairs of equations until an equation with a single variale remains. Our plan of attack on this system will e to use equation to eliminate the variale x from equations and. We egin y multiplying each side of equation y -4 and adding the result to equation. (Do you see why? The coefficients of x are now negatives of one another.) We also multiply equation y -2 and add the result to equation. Notice that these two procedures result in the removal of the variale x from equations and. x y z 1 4x 3y 2z 16 x y z 1 2x 2y 3z 5 Multiply y 4 Multiply y 2 4x 4y 4z 4 4x 3y 2z 16 7y 6z 20 Add. 2x 2y 2z 2 2x 2y 3z 5 4y z 7 Add. x y z 1 7y 6z 20 4y z 7 We now concentrate on equations and, treating them as a system of two equations containing two variales. It is easiest to eliminate z. We multiply each side of equation y 6 and add equations and. The result is the new equation. 7y 6z 20 7y 6z 20 4y z 7 Multiply y 6 24y 6z 42 31y 62 Add. x y z 1 7y 6z 20 31y 62

10 SECTION 10.1 Systems of Linear Equations: Sustitution and Elimination 735 We now solve equation for y y dividing oth sides of the equation y -31. x + y - z = c -7y + 6z = y = Back-sustitute y = -2 in equation and solve for z. -7y + 6z = z = 20 6z = 6 z = 1 Sustitute y = -2 in. Sutract 14 from oth sides of the equation. Divide oth sides of the equation y 6. Finally, we ack-sustitute y = -2 and z = 1 in equation and solve for x. x + y - z = -1 x = -1 x - 3 = -1 x = 2 Sustitute y = -2 and z = 1 in. Simplify. Add 3 to oth sides. The solution of the original system is x = 2, y = -2, z = 1. You should verify this solution. Look ack over the solution given in Example 9. Note the pattern of removing one of the variales from two of the equations, followed y solving this system of two equations and two unknowns. Although which variales to remove is your choice, the methodology remains the same for all systems. NOW WORK PROBLEM Identify Inconsistent Systems of Equations Containing Three Variales EXAMPLE 10 An Inconsistent System of Linear Equations 2x + y - z = -2 c x + 2y - z = -9 x - 4y + z = 1 Our plan of attack is the same as in Example 9. However, in this system, it seems easiest to eliminate the variale z first. Do you see why? Multiply each side of equation y -1 and add the result to equation.add equations and. 2x y z 2 x 2y z 9 x y 7 x 2y z 9 x 4y + z 1 2x 2y 8 Multiply y 1. Add. Add. 2x y z 2 x y 7 2x 2y 8

11 736 CHAPTER 10 Systems of Equations and Inequalities x y 7 2x 2y 8 We now concentrate on equations and, treating them as a system of two equations containing two variales. Multiply each side of equation y 2 and add the result to equation. Multiply y 2. 2x 2y 14 2x 2y Add. Equation has no solution and the system is inconsistent. 2x y z 2 x y EXAMPLE 11 7 Express the of a System of Dependent Equations Containing Three Variales Solving a System of Dependent Equations x - 2y - z = 8 c 2x - 3y + z = 23 4x - 5y + 5z = 53 x 2y z 8 2x 3y z 23 x 2y z 8 4x 5y 5z 53 Our plan is to eliminate x from equations and. Multiply each side of equation y -2 and add the result to equation.also, multiply each side of equation y -4 and add the result to equation. Multiply y 2. 2x 4y 2z 16 2x 3y z 23 y 3z 7 Multiply y 4. 4x 8y 4z 32 4x 5y 5z 53 3y 9z 21 Add. Treat equations and as a system of two equations containing two variales, and eliminate the variale y y multiplying oth sides of equation y -3 and adding the result to equation. y 3z 7 Multiply y 3. 3y 9z 21 x 2y z 8 3y 9z 21 3y 9z 21 Add. y 3z The original system is equivalent to a system containing two equations, so the equations are dependent and the system has infinitely many solutions. If we solve equation for y, we can express y in terms of z as y = -3z + 7. Sustitute this expression into equation to determine x in terms of z. x z z = 8 x + 6z z = 8 We will write the solution to the system as where z can e any real numer. x - 2y - z = 8 x + 5z = 22 Add. x = -5z + 22 x = -5z + 22 e y = -3z + 7 x 2y z 8 y 3z 7 3y 9z 21 Sustitute y = -3z + 7 in. Remove parentheses. Comine like terms. Solve for x.

12 SECTION 10.1 Systems of Linear Equations: Sustitution and Elimination 737 This way of writing the solution makes it easier to find specific solutions of the system. To find specific solutions, choose any value of z and use the equations x = -5z + 22 and y = -3z + 7 to determine x and y. For example, if z = 0, then x = 22 and y = 7, and if z = 1, then x = 17 and y = 4. NOW WORK PROBLEM 45. Two points in the Cartesian plane determine a unique line. Given three noncollinear points, we can find the (unique) quadratic function whose graph contains these three points. EXAMPLE 12 Curve Fitting Find real numers a,, and c so that the graph of the quadratic function y = ax 2 + x + c contains the points 1-1, -42, 11, 62, and 13, 02. We require that the three points satisfy the equation y = ax 2 + x + c. For the point 1-1, -42 we have: -4 = a c -4 = a - + c For the point 11, 62 we have: 6 = a c 6 = a + + c For the point 13, 02 we have: 0 = a c 0 = 9a c We wish to determine a,, and c so that each equation is satisfied. That is, we want to solve the following system of three equations containing three variales: Figure a - + c = -4 c a + + c = 6 9a c = 0 Solving this system of equations, we otain a = -2, = 5, and c = 3. So the quadratic function whose graph contains the points 1-1, -42, 11, 62, and 13, 02 is y = -2x 2 + 5x + 3 y = ax 2 + x + c, a = -2, = 5, c = 3 Figure 6 shows the graph of the function along with the three points.

### 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 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

### A synonym is a word that has the same or almost the same definition of

Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### 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

### 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

### 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,

### 2013 MBA Jump Start Program

2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

### Systems of Linear Equations

Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11-]. Main points in this section: 1. Definition of Linear

### Non-Linear Regression 2006-2008 Samuel L. Baker

NON-LINEAR REGRESSION 1 Non-Linear Regression 2006-2008 Samuel L. Baker The linear least squares method that you have een using fits a straight line or a flat plane to a unch of data points. Sometimes

### 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

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

### Answers Teacher Copy. Systems of Linear Equations Monetary Systems Overload. Activity 3. Solving Systems of Two Equations in Two Variables

of 26 8/20/2014 2:00 PM Answers Teacher Copy Activity 3 Lesson 3-1 Systems of Linear Equations Monetary Systems Overload Solving Systems of Two Equations in Two Variables Plan Pacing: 1 class period Chunking

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### 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

### 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

### 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

### BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

### Answer Key Building Polynomial Functions

Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,

### 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

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

### Chesapeake College. Syllabus. MAT 031 Elementary Algebra (silcox87517) John H. Silcox

Chesapeake College Syllabus MAT 031 Elementary Algebra (silcox87517) John H. Silcox Phone 410-778-4540 (W) Phone 410-348-5370 (H) john.silcox@1972.usna.com Syllabus For MAT 031 I. OFFICE HOURS: One hour

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Algebra Bridge Project Cell Phone Plans

Algebra Bridge Project Cell Phone Plans Name Teacher Part I: Two Cell Phone Plans You are in the market for a new cell phone, and you have narrowed your search to two different cell phone companies --

### North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Community College System Diagnostic and Placement Test Sample Questions 01 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### 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

### Solving Quadratic & Higher Degree Inequalities

Ch. 8 Solving Quadratic & Higher Degree Inequalities We solve quadratic and higher degree inequalities very much like we solve quadratic and higher degree equations. One method we often use to solve quadratic

### Solving Quadratic Equations by Factoring

4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written

### http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

### 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

### 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

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### GRADES 7, 8, AND 9 BIG IDEAS

Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for

### 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

### 9 Multiplication of Vectors: The Scalar or Dot Product

Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

### Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction

Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a

### Midterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013

Midterm Review Problems (the first 7 pages) Math 1-5116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in

### Systems of Linear Equations; Matrices

4 Systems of Linear Equations; Matrices 4. Review: Systems of Linear Equations in Two Variables 4. Systems of Linear Equations and Augmented Matrices 4. Gauss Jordan Elimination 4.4 Matrices: Basic Operations

### Handout for three day Learning Curve Workshop

Handout for three day Learning Curve Workshop Unit and Cumulative Average Formulations DAUMW (Credits to Professors Steve Malashevitz, Bo Williams, and prior faculty. Blame to Dr. Roland Kankey, roland.kankey@dau.mil)

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014))

SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014)) There are so far 8 most common methods to solve quadratic equations in standard form ax² + bx + c = 0.

### CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

### Lesson 9: Radicals and Conjugates

Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.

### Overview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: Slope-Intercept Form

Name Date Linear Functions: Slope-Intercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review,

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Homework #1 Solutions

Homework #1 Solutions Problems Section 1.1: 8, 10, 12, 14, 16 Section 1.2: 2, 8, 10, 12, 16, 24, 26 Extra Problems #1 and #2 1.1.8. Find f (5) if f (x) = 10x x 2. Solution: Setting x = 5, f (5) = 10(5)

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

### GRADE 8 MATH: TALK AND TEXT PLANS

GRADE 8 MATH: TALK AND TEXT PLANS UNIT OVERVIEW This packet contains a curriculum-embedded Common Core standards aligned task and instructional supports. The task is embedded in a three week unit on systems

### Some Lecture Notes and In-Class Examples for Pre-Calculus:

Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax

### SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

### Notes from February 11

Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### 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

### DRAFT. Algebra 1 EOC Item Specifications

DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

### ACCUPLACER. Testing & Study Guide. Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011

ACCUPLACER Testing & Study Guide Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011 Thank you to Johnston Community College staff for giving permission to revise

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### Factoring Quadratic Expressions

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### Florida Math for College Readiness

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

### Arithmetic and Algebra of Matrices

Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational

### 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

### Transition To College Mathematics

Transition To College Mathematics In Support of Kentucky s College and Career Readiness Program Northern Kentucky University Kentucky Online Testing (KYOTE) Group Steve Newman Mike Waters Janis Broering

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### Balanced Assessment Test Algebra 2008

Balanced Assessment Test Algebra 2008 Core Idea Task Score Representations Expressions This task asks students find algebraic expressions for area and perimeter of parallelograms and trapezoids. Successful

### 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,

### 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,

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### Prerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.

Course Syllabus Math 1314 College Algebra Revision Date: 8-21-15 Catalog Description: In-depth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems

### Lecture 2: August 29. Linear Programming (part I)

10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.

### Algebra II. Weeks 1-3 TEKS

Algebra II Pacing Guide Weeks 1-3: Equations and Inequalities: Solve Linear Equations, Solve Linear Inequalities, Solve Absolute Value Equations and Inequalities. Weeks 4-6: Linear Equations and Functions:

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Equations of Lines and Planes

Calculus 3 Lia Vas Equations of Lines and Planes Planes. A plane is uniquely determined by a point in it and a vector perpendicular to it. An equation of the plane passing the point (x 0, y 0, z 0 ) perpendicular

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### COLLEGE ALGEBRA 10 TH EDITION LIAL HORNSBY SCHNEIDER 1.1-1

10 TH EDITION COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER 1.1-1 1.1 Linear Equations Basic Terminology of Equations Solving Linear Equations Identities 1.1-2 Equations An equation is a statement that two expressions

### Introduction to Quadratic Functions

Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

### 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

### When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.

Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

### Administrative - Master Syllabus COVER SHEET

Administrative - Master Syllabus COVER SHEET Purpose: It is the intention of this to provide a general description of the course, outline the required elements of the course and to lay the foundation for

### Algebra 1 End-of-Course Exam Practice Test with Solutions

Algebra 1 End-of-Course Exam Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fill-in Response Items, write your answer in the box provided, placing one digit in

### In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

The vector product mc-ty-vectorprod-2009-1 Oneofthewaysinwhichtwovectorscanecominedisknownasthevectorproduct.When wecalculatethevectorproductoftwovectorstheresult,asthenamesuggests,isavector. Inthisunityouwilllearnhowtocalculatethevectorproductandmeetsomegeometricalapplications.

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### 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

### SUFFOLK COMMUNITY COLLEGE MATHEMATICS AND COMPUTER SCIENCE DEPARTMENT STUDENT COURSE OUTLINE Fall 2011

SUFFOLK COMMUNITY COLLEGE MATHEMATICS AND COMPUTER SCIENCE DEPARTMENT STUDENT COURSE OUTLINE Fall 2011 INSTRUCTOR: Professor Emeritus Donald R. Coscia OFFICE: Online COURSE: MAT111-Algebra II SECTION &

### 1.5 Equations of Lines and Planes in 3-D

40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from

### CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA. Semester Hours Credit: 3

CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA Semester Hours Credit: 3 (This course is equivalent to DSMA 0301. The difference being that this course is offered only on those campuses