Systems of Linear Equations: Solving by Substitution

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Systems of Linear Equations: Solving by Substitution"

Transcription

1 8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing and addition as methods of solving linear sstems. A third method is called solution b substitution. Eample 1 Solving a Sstem b Substitution Solve b substitution. 12 (1) 3 (2) Notice that equation (2) sas that and 3 name the same quantit. So we ma substitute 3 for in equation (1). We then have Replace with 3 in equation (1). NOTE The resulting equation contains onl the variable, so substitution is just another wa of eliminating one of the variables from our sstem. NOTE The solution for a sstem is written as an ordered pair We can now substitute 3 for in equation (1) to find the corresponding coordinate of the solution So (3, 9) is the solution. This last step is identical to the one ou saw in Section 8.2. As before, ou can substitute the known coordinate value back into either of the original equations to find the value of the remaining variable. The check is also identical. CHECK YOURSELF 1 Solve b substitution. 9 4 The same technique can be readil used an time one of the equations is alread solved for or for, as Eample 2 illustrates. 657

2 658 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS Eample 2 Solving a Sstem b Substitution Solve b substitution (1) 2 7 (2) Because equation (2) tells us that is 2 7, we can replace with 2 7 in equation (1). This gives NOTE Now is eliminated from the equation, and we can proceed to solve for. 2 3(2 7) We now know that 3 is the coordinate for the solution. So substituting 3 for in equation (2), we have And (3, 1) is the solution. Once again ou should verif this result b letting 3 and 1 in the original sstem. CHECK YOURSELF 2 Solve b substitution As we have seen, the substitution method works ver well when one of the given equations is alread solved for or for. It is also useful if ou can readil solve for or for in one of the equations. Eample 3 Solving a Sstem b Substitution Solve b substitution. 2 5 (1) 3 8 (2)

3 SYSTEMS OF LINEAR EQUATIONS: SOLVING BY SUBSTITUTION SECTION Neither equation is solved for a variable. That is easil handled in this case. Solving for in equation (1), we have NOTE Equation (2) could have been solved for with the result substituted into equation (1). 2 5 Now substitute 2 5 for in equation (2). 3(2 5) Substituting 1 for in equation (2) ields 3 ( 1) So (3, 1) is the solution. You should check this result b substituting 3 for and 1 for in the equations of the original sstem. CHECK YOURSELF 3 Solve b substitution Inconsistent sstems and dependent sstems will show up in a fashion similar to that which we saw in Section 8.2. Eample 4 illustrates this approach. Eample 4 Solving an Inconsistent or Dependent Sstem Solve the following sstems b substitution. (a) (1) 2 3 (2) NOTE Don t forget to change both signs in the parentheses. From equation (2) we can substitute 2 3 for in equation (1). 4 2(2 3) Both variables have been eliminated, and we have the true statement 6 6.

4 660 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS Recall from the last section that a true statement tells us that the lines coincide. We call this sstem dependent. There are an infinite number of solutions. (b) (3) 2 2 (4) Substitute 2 2 for in equation (3). 3(2 2) This time we have a false statement. This means that the sstem is inconsistent and that the graphs of the two equations are parallel lines. There is no solution. CHECK YOURSELF 4 Indicate whether the sstems are inconsistent (no solution) or dependent (an infinite number of solutions). (a) (b) The following summarizes our work in this section. Step b Step: To Solve a Sstem of Linear Equations b Substitution Step 1 Step 2 Step 3 Step 4 Step 5 Solve one of the given equations for or. If this is alread done, go on to step 2. Substitute this epression for or for into the other equation. Solve the resulting equation for the remaining variable. Substitute the known value into either of the original equations to find the value of the second variable. Check our solution in both of the original equations. You have now seen three different was to solve sstems of linear equations: b graphing, adding, and substitution. The natural question is, Which method should I use in a given situation? Graphing is the least eact of the methods, and solutions ma have to be estimated. The algebraic methods addition and substitution give eact solutions, and both will work for an sstem of linear equations. In fact, ou ma have noticed that several eamples in this section could just as easil have been solved b adding (Eample 3, for instance). The choice of which algebraic method (substitution or addition) to use is ours and depends largel on the given sstem. Here are some guidelines designed to help ou choose an appropriate method for solving a linear sstem.

5 SYSTEMS OF LINEAR EQUATIONS: SOLVING BY SUBSTITUTION SECTION Rules and Properties: Choosing an Appropriate Method for Solving a Sstem 1. If one of the equations is alread solved for (or for ), then substitution is the preferred method. 2. If the coefficients of (or of ) are the same, or opposites, in the two equations, then addition is the preferred method. 3. If solving for (or for ) in either of the given equations will result in fractional coefficients, then addition is the preferred method. Eample 5 Choosing an Appropriate Method for Solving a Sstem Select the most appropriate method for solving each of the following sstems. (a) Addition is the most appropriate method because solving for a variable will result in fractional coefficients. (b) Substitution is the most appropriate method because the second equation is alread solved for. (c) Addition is the most appropriate method because the coefficients of are opposites. CHECK YOURSELF 5 Select the most appropriate method for solving each of the following sstems. (a) (b) (c) (d) Number problems, such as those presented in Chapter 2, are sometimes more easil solved b the methods presented in this section. Eample 6 illustrates this approach. Eample 6 Solving a Number Problem b Substitution The sum of two numbers is 25. If the second number is 5 less than twice the first number, what are the two numbers?

6 662 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS NOTE 1. What do ou want to find? 2. Assign variables. This time we use two letters, and. 3. Write equations for the solution. Here two equations are needed because we have introduced two variables. Step 1 Step 2 Step 3 25 The sum is You want to find the two unknown numbers. Let the first number and the second number. The second number is 5 less than twice the first. 4. Solve the sstem of equations. NOTE We use the substitution method because equation (2) is alread solved for. 5. Check the result. Step 4 25 (1) 2 5 (2) Substitute 2 5 for in equation (1). (2 5) From equation (1), The two numbers are 10 and 15. Step 5 The sum of the numbers is 25. The second number, 15, is 5 less than twice the first number, 10. The solution checks. CHECK YOURSELF 6 The sum of two numbers is 28. The second number is 4 more than twice the first number. What are the numbers? Sketches are alwas helpful in solving applications from geometr. Let s look at such an eample. Eample 7 Solving an Application from Geometr The length of a rectangle is 3 meters (m) more than twice its width. If the perimeter of the rectangle is 42 m, find the dimensions of the rectangle. Step 1 You want to find the dimensions (length and width) of the rectangle.

7 SYSTEMS OF LINEAR EQUATIONS: SOLVING BY SUBSTITUTION SECTION NOTE We used and as our two variables in the previous eamples. Use whatever letters ou want. The process is the same, and sometimes it helps ou remember what letter stands for what. Here L length and W width. Step 2 problem. W Let L be the length of the rectangle and W the width. Now draw a sketch of the L W Step 3 L Write the equations for the solution. L 2W 3 3 more than twice the width 2L 2W 42 The perimeter Step 4 Solve the sstem. NOTE Substitution is used because one equation is alread solved for a variable. L 2W 3 (1) 2L 2W 42 (2) From equation (1) we can substitute 2W 3 for L in equation (2). 2(2W 3) 2W 42 4W 6 2W 42 6W 36 W 6 Replace W with 6 in equation (1) to find L. L The length is 15 m, the width is 6 m. 6 m 15 m Step 5 Check these results. The perimeter is 2L 2W, which should give us 42 m. 2(15) 2(6) CHECK YOURSELF 7 The length of each of the two equal legs of an isosceles triangle is 5 in. less than the length of the base. If the perimeter of the triangle is 50 in., find the lengths of the legs and the base.

8 664 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS CHECK YOURSELF ANSWERS 1. ( 3, 12) 2. (6, 2) 3. (2, 1) 4. (a) Inconsistent sstem; (b) Dependent sstem 5. (a) Addition; (b) Substitution; (c) Substitution; (d) Addition 6. The numbers are 8 and The legs have length 15 in.; the base is 20 in.

9 Name 8.3 Eercises Section Date Solve each of the following sstems b substitution ANSWERS

10 ANSWERS Solve each of the following sstems b using either addition or substitution. If a unique solution does not eist, state whether the sstem is dependent or inconsistent Solve each sstem

11 ANSWERS Solve each of the following problems. Be sure to show the equation used for the solution. 47. Number problem. The sum of two numbers is 100. The second is three times the first. Find the two numbers Number problem. The sum of two numbers is 70. The second is 10 more than 3 times the first. Find the numbers. 49. Number problem. The sum of two numbers is 56. The second is 4 less than twice the first. What are the two numbers? 50. Number problem. The difference of two numbers is 4. The larger is 8 less than twice the smaller. What are the two numbers? Number problem. The difference of two numbers is 22. The larger is 2 more than 3 times the smaller. Find the two numbers. 52. Number problem. One number is 18 more than another, and the sum of the smaller number and twice the larger number is 45. Find the two numbers. 53. Number problem. One number is 5 times another. The larger number is 9 more than twice the smaller. Find the two numbers. 54. Package weight. Two packages together weigh 32 kilograms (kg). The smaller package weighs 6 kg less than the larger. How much does each package weigh? 55. Appliance costs. A washer-drer combination costs $1200. If the washer costs $220 more than the drer, what does each appliance cost separatel? 56. Voting trends. In a town election, the winning candidate had 220 more votes than the loser. If 810 votes were cast in all, how man votes did each candidate receive? 667

12 ANSWERS Cost of furniture. An office desk and chair together cost $850. If the desk cost $50 less than twice as much as the chair, what did each cost? a. b. 58. Dimensions of a rectangle. The length of a rectangle is 2 inches (in.) more than twice its width. If the perimeter of the rectangle is 34 in., find the dimensions of the rectangle. 59. Perimeter. The perimeter of an isosceles triangle is 37 in. The lengths of the two equal legs are 6 in. less than 3 times the length of the base. Find the lengths of the three sides. 60. You have a part-time job writing the Consumer Concerns column for our local newspaper. Your topic for this week is clothes drers, and ou are planning to compare the Helpmate and the Whirlgarb drers, both readil available in stores in our area. The information ou have is that the Helpmate drer is listed at $520, and it costs 22.5 to dr an average size load at the utilit rates in our cit. The Whirlgarb drer is listed at $735, and it costs 15.8 to run for each normal load. The maintenance costs for both drers are about the same. Working with a partner, write a short article giving our readers helpful advice about these appliances. What should the consider when buing one of these clothes drers? Getting Read for Section 8.4 [Section 2.7] Graph the solution sets for the following linear inequalities. (a) 8 (b)

13 ANSWERS (c) (d) 2 c. d. e. f. (e) 3 (f) 5 Answers 1. (2, 8) 3. (4, 2) (4, 1) 9. (10, 6) 11. (5, 3) 13. (10, 2) No solution 19. (3, 3) (4, 1) 25. Infinite number of solutions 27. (10, 1) 29. ( 2, 1) 31. (0, 2) 33. (2, 3) 35. Dependent sstem , 3 1 3, 2 3 2, 3 4 3, (0, 10) 45. (2, 1) , , , , Washer $710, drer $ Desk $550, chair $ in., 15 in., 15 in. a. 8 b , , 2 669

14 c d. 2 e. 3 f

7.3 Solving Systems by Elimination

7.3 Solving Systems by Elimination 7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need

More information

SYSTEMS OF LINEAR EQUATIONS

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

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

The Graph of a Linear Equation

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

More information

Graphing Nonlinear Systems

Graphing Nonlinear Systems 10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities

More information

Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

More information

Graphing Linear Equations

Graphing Linear Equations 6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

More information

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

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

More information

MULTIPLE REPRESENTATIONS through 4.1.7

MULTIPLE REPRESENTATIONS through 4.1.7 MULTIPLE REPRESENTATIONS 4.1.1 through 4.1.7 The first part of Chapter 4 ties together several was to represent the same relationship. The basis for an relationship is a consistent pattern that connects

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

Systems of Equations. from Campus to Careers Fashion Designer

Systems of Equations. from Campus to Careers Fashion Designer Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.

More information

Graphing Linear Inequalities in Two Variables

Graphing Linear Inequalities in Two Variables 5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

8.7 Systems of Non-Linear Equations and Inequalities

8.7 Systems of Non-Linear Equations and Inequalities 8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

Solving Special Systems of Linear Equations

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

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations 5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

More information

Simplification of Rational Expressions and Functions

Simplification of Rational Expressions and Functions 7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work

More information

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0 LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

More information

SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION 2.2. Distance and Midpoint Formulas; Circles SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

More information

We start with the basic operations on polynomials, that is adding, subtracting, and multiplying.

We start with the basic operations on polynomials, that is adding, subtracting, and multiplying. R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract

More information

Unit 1 Study Guide Systems of Linear Equations and Inequalities. Part 1: Determine if an ordered pair is a solution to a system

Unit 1 Study Guide Systems of Linear Equations and Inequalities. Part 1: Determine if an ordered pair is a solution to a system Unit Stud Guide Sstems of Linear Equations and Inequalities 6- Solving Sstems b Graphing Part : Determine if an ordered pair is a solution to a sstem e: (, ) Eercises: substitute in for and - in for in

More information

Rectangle Square Triangle

Rectangle Square Triangle HFCC Math Lab Beginning Algebra - 15 PERIMETER WORD PROBLEMS The perimeter of a plane geometric figure is the sum of the lengths of its sides. In this handout, we will deal with perimeter problems involving

More information

To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

More information

Chapter 3 & 8.1-8.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x - 4y = 10 16x + 8y = 10

Chapter 3 & 8.1-8.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x - 4y = 10 16x + 8y = 10 Chapter 3 & 8.1-8.3 These are meant for practice. The actual test is different. Determine whether the pair of equations represents parallel lines. 1) 9 + 3 = 12 27 + 9 = 39 1) Determine whether the pair

More information

{ } Sec 3.1 Systems of Linear Equations in Two Variables

{ } Sec 3.1 Systems of Linear Equations in Two Variables Sec.1 Sstems of Linear Equations in Two Variables Learning Objectives: 1. Deciding whether an ordered pair is a solution.. Solve a sstem of linear equations using the graphing, substitution, and elimination

More information

The Quadratic Function

The Quadratic Function 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

More information

Example 1: Model A Model B Total Available. Gizmos. Dodads. System:

Example 1: Model A Model B Total Available. Gizmos. Dodads. System: Lesson : Sstems of Equations and Matrices Outline Objectives: I can solve sstems of three linear equations in three variables. I can solve sstems of linear inequalities I can model and solve real-world

More information

LINEAR FUNCTIONS OF 2 VARIABLES

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

More information

Introduction. Introduction

Introduction. Introduction Introduction Solving Sstems of Equations Let s start with an eample. Recall the application of sales forecasting from the Working with Linear Equations module. We used historical data to derive the equation

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations 5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

More information

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system. _.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

More information

Coordinate Geometry. Positive gradients: Negative gradients:

Coordinate Geometry. Positive gradients: Negative gradients: 8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight

More information

Higher. Polynomials and Quadratics 64

Higher. Polynomials and Quadratics 64 hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

More information

Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.

Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button. Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.

More information

Anytime plan TalkMore plan

Anytime plan TalkMore plan CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations

More information

Equations Involving Fractions

Equations Involving Fractions . Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Right Triangles and SOHCAHTOA: Finding the Length of a Side Given One Side and One Angle

Right Triangles and SOHCAHTOA: Finding the Length of a Side Given One Side and One Angle Right Triangles and SOHCAHTOA: Finding the Length of a Side Given One Side and One Angle Preliminar Information: is an acronm to represent the following three trigonometric ratios or formulas: opposite

More information

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets

More information

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions: Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

More information

2.3 Quadratic Functions

2.3 Quadratic Functions . Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the

More information

Essential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars)

Essential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars) 5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bed-and-breakfast.

More information

5.2 Inverse Functions

5.2 Inverse Functions 78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

More information

Linear Inequalities, Systems, and Linear Programming

Linear Inequalities, Systems, and Linear Programming 8.8 Linear Inequalities, Sstems, and Linear Programming 481 8.8 Linear Inequalities, Sstems, and Linear Programming Linear Inequalities in Two Variables Linear inequalities with one variable were graphed

More information

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES 6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

More information

More Equations and Inequalities

More Equations and Inequalities Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

More information

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

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

More information

GRAPHING SYSTEMS OF LINEAR INEQUALITIES

GRAPHING SYSTEMS OF LINEAR INEQUALITIES 444 (8 5) Chapter 8 Sstems of Linear Equations and Inequalities GETTING MORE INVOLVED 5. Discussion. When asked to graph the inequalit, a student found that (0, 5) and (8, 0) both satisfied. The student

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

Solving Systems of Equations

Solving Systems of Equations Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that

More information

Solve the linear programming problem graphically: Minimize w 4. subject to. on the vertical axis.

Solve the linear programming problem graphically: Minimize w 4. subject to. on the vertical axis. Do a similar example with checks along the wa to insure student can find each corner point, fill out the table, and pick the optimal value. Example 3 Solve the Linear Programming Problem Graphicall Solve

More information

Solving Equations by the Multiplication Property

Solving Equations by the Multiplication Property 2.2 Solving Equations by the Multiplication Property 2.2 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the multiplication property to solve equations. Find the mean

More information

Systems of Linear Equations in Three Variables

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

More information

Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8. Lines and Planes. By the end of this chapter, you will Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

More information

Q (x 1, y 1 ) m = y 1 y 0

Q (x 1, y 1 ) m = y 1 y 0 . Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine

More information

Filling in Coordinate Grid Planes

Filling in Coordinate Grid Planes Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the

More information

7.3 Graphing Rational Functions

7.3 Graphing Rational Functions Section 7.3 Graphing Rational Functions 639 7.3 Graphing Rational Functions We ve seen that the denominator of a rational function is never allowed to equal zero; division b zero is not defined. So, with

More information

Section 7.1 Graphing Linear Inequalities in Two Variables

Section 7.1 Graphing Linear Inequalities in Two Variables Section 7.1 Graphing Linear Inequalities in Two Variables Eamples of linear inequalities in two variables include + 6, and 1 A solution of a linear inequalit is an ordered pair that satisfies the

More information

10.2 The Unit Circle: Cosine and Sine

10.2 The Unit Circle: Cosine and Sine 0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on

More information

ALGEBRA 1 SKILL BUILDERS

ALGEBRA 1 SKILL BUILDERS ALGEBRA 1 SKILL BUILDERS (Etra Practice) Introduction to Students and Their Teachers Learning is an individual endeavor. Some ideas come easil; others take time--sometimes lots of time- -to grasp. In addition,

More information

Introduction to the Practice Exams

Introduction to the Practice Exams Introduction to the Practice Eams The math placement eam determines what math course you will start with at North Hennepin Community College. The placement eam starts with a 1 question elementary algebra

More information

Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.2: Magnitudes, Directions, and Components of Vectors Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

More information

The majority of college students hold credit cards. According to the Nellie May

The majority of college students hold credit cards. According to the Nellie May CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials

More information

3.1 Graphically Solving Systems of Two Equations

3.1 Graphically Solving Systems of Two Equations 3.1 Graphicall Solving Sstems of Two Equations (Page 1 of 24) 3.1 Graphicall Solving Sstems of Two Equations Definitions The plot of all points that satisf an equation forms the graph of the equation.

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

1.2 GRAPHS OF EQUATIONS

1.2 GRAPHS OF EQUATIONS 000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

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

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

More information

2.4 Inequalities with Absolute Value and Quadratic Functions

2.4 Inequalities with Absolute Value and Quadratic Functions 08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we

More information

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

More information

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

More information

Linear Inequality in Two Variables

Linear Inequality in Two Variables 90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.

More information

Essential Question: What are two ways to solve an absolute value inequality? A2.6.F Solve absolute value linear inequalities.

Essential Question: What are two ways to solve an absolute value inequality? A2.6.F Solve absolute value linear inequalities. Locker LESSON.3 Solving Absolute Value Inequalities Name Class Date.3 Solving Absolute Value Inequalities Teas Math Standards The student is epected to: A.6.F Essential Question: What are two was to solve

More information

11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems

11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems a b Objectives Solve maimum minimum problems involving quadratic functions. Fit a quadratic function to a set of data to form a mathematical model, and solve related applied problems. 11.7 MATHEMATICAL

More information

Systems of Linear Equations - Introduction

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

More information

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

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

More information

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear

More information

1.6 Graphs of Functions

1.6 Graphs of Functions .6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each -coordinate was matched with onl one -coordinate. We spent most of our time

More information

MAT 080-Algebra II Applications of Quadratic Equations

MAT 080-Algebra II Applications of Quadratic Equations MAT 080-Algebra II Applications of Quadratic Equations Objectives a Applications involving rectangles b Applications involving right triangles a Applications involving rectangles One of the common applications

More information

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right

More information

P1. Plot the following points on the real. P2. Determine which of the following are solutions

P1. Plot the following points on the real. P2. Determine which of the following are solutions Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in

More information

COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS

COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs

More information

Skill Builders. (Extra Practice) Volume I

Skill Builders. (Extra Practice) Volume I Skill Builders (Etra Practice) Volume I 1. Factoring Out Monomial Terms. Laws of Eponents 3. Function Notation 4. Properties of Lines 5. Multiplying Binomials 6. Special Triangles 7. Simplifying and Combining

More information

Formulas and Problem Solving

Formulas and Problem Solving 2.4 Formulas and Problem Solving 2.4 OBJECTIVES. Solve a literal equation for one of its variables 2. Translate a word statement to an equation 3. Use an equation to solve an application Formulas are extremely

More information

Chapter 2: Boolean Algebra and Logic Gates. Boolean Algebra

Chapter 2: Boolean Algebra and Logic Gates. Boolean Algebra The Universit Of Alabama in Huntsville Computer Science Chapter 2: Boolean Algebra and Logic Gates The Universit Of Alabama in Huntsville Computer Science Boolean Algebra The algebraic sstem usuall used

More information

Are You Ready? Simplify Radical Expressions

Are You Ready? Simplify Radical Expressions SKILL Are You Read? Simplif Radical Epressions Teaching Skill Objective Simplif radical epressions. Review with students the definition of simplest form. Ask: Is written in simplest form? (No) Wh or wh

More information

9.3 OPERATIONS WITH RADICALS

9.3 OPERATIONS WITH RADICALS 9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in

More information

Characteristics of the Four Main Geometrical Figures

Characteristics of the Four Main Geometrical Figures Math 40 9.7 & 9.8: The Big Four Square, Rectangle, Triangle, Circle Pre Algebra We will be focusing our attention on the formulas for the area and perimeter of a square, rectangle, triangle, and a circle.

More information

Math 40 Chapter 3 Lecture Notes. Professor Miguel Ornelas

Math 40 Chapter 3 Lecture Notes. Professor Miguel Ornelas Math 0 Chapter Lecture Notes Professor Miguel Ornelas M. Ornelas Math 0 Lecture Notes Section. Section. The Rectangular Coordinate Sstem Plot each ordered pair on a Rectangular Coordinate Sstem and name

More information

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular

More information

3.4 The Point-Slope Form of a Line

3.4 The Point-Slope Form of a Line Section 3.4 The Point-Slope Form of a Line 293 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( = m + b). The slope-intercept form of a line is

More information

Some Tools for Teaching Mathematical Literacy

Some Tools for Teaching Mathematical Literacy Some Tools for Teaching Mathematical Literac Julie Learned, Universit of Michigan Januar 200. Reading Mathematical Word Problems 2. Fraer Model of Concept Development 3. Building Mathematical Vocabular

More information

EQUATIONS. Main Overarching Questions: 1. What is a variable and what does it represent?

EQUATIONS. Main Overarching Questions: 1. What is a variable and what does it represent? EQUATIONS Introduction to Variables, Algebraic Expressions, and Equations (2 days) Overview of Objectives, students should be able to: Main Overarching Questions: 1. Evaluate algebraic expressions given

More information

8 Graphs of Quadratic Expressions: The Parabola

8 Graphs of Quadratic Expressions: The Parabola 8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be

More information

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m 0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

More information