Name Period Date LINEAR FUNCTIONS STUDENT PACKET 6: SYSTEMS OF LINEAR EQUATIONS LF6.1 LF6.2 LF6.3 Introduction to Systems of Linear Equations Understand the definition of a system of linear equations Understand the definition of a solution to a system of equations Know the different types of solutions possible Solve systems of equations using the graphing method Solving Systems Using Algebra Solve systems of linear equations using tables, graphs, and algebraic notation. Compare the effectiveness of different strategies for solving systems of equations. Understand and be able to use the substitution and elimination methods for solving systems. Use the addition property of equality, the multiplication property of equality, and the substitution property to manipulate expressions and equations Solving Systems of Linear Inequalities Solve systems of linear inequalities by graphing Use the fourfold way to show complete solutions to a complex problem LF6 STUDENT PAGES LF6.4 Vocabulary, Skill Builders, and Review 26 1 7 21 LF6 SP
s WORD BANK Word Phrase Definition or Explanation Example or Picture dependent system of equations inconsistent system of equations independent system of equations slope of a line solution of a system of equations system of linear equations x-intercept y-intercept LF6 SP0
6.1 Introduction to Systems of Linear Equations INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS Ready (Summary) We will learn about systems of linear equations and how to solve them by graphing. We will use algebra to write linear equations in equivalent forms. Go (Warmup) Set (Goals) Understand the definition of a system of linear equations Understand the definition of a solution to a system of equations Know the different types of solutions possible Solve systems of equations using the graphing method Graph lines that fit the descriptions. Find the slope, the y-intercept, and the equation of each line in slope-intercept form. 1. The line that goes through the point (-2, 1) and has a slope of -3. slope: y-intercept: equation: y = 2. The line that goes through the points (3, 6) and (-3, 8). slope: y-intercept: equation: y = LF6 SP1
6.1 Introduction to Systems of Linear Equations CHANGING EQUATIONS TO SLOPE-INTERCEPT FORM In previous lessons you learned about the slope-intercept form of a line, y = mx + b. We say that for this equation y is written in terms of x. (The coefficient of y must be +1). 1. What is advantageous about having an equation in this form? Describe how knowing the values of m and b help with graphing a line quickly. Write each equation in slope intercept form. If the equation is already in slope-intercept form, then circle the equation. 2. 4x + 2y + 12 = 0 3. 6x y = -7 4. 3x 17 = y 5. -y = 2x + 5 6. 1 2 y = 5x 1 7. x + 2 3 y = -6 LF6 SP2
6.1 Introduction to Systems of Linear Equations SAVING FOR A PRINTER REVISITED Theresa and Cary are saving for a printer. Theresa has $10 in the bank and will save $20 each month. Cary has $25 in the bank and will save $15 each month 1. How much does Theresa have at month #0? 2. How much does Cary have at month #0? 3. Fill in the table for several months. Then graph the data. Use a different color for each girl s table values and graph. Label and scale the axes appropriately. Month # Theresa Month # (x) (y) (x) 0 0 4. Write an equation that represents Theresa s savings. y = 5. Write is an equation that represents Cary s savings. y = 1 1 2 2 3 3 4 4 5 5 6 6 7 7 Cary (y) 6. Which girl is saving at a faster rate? Justify your answer by referring to the table, the graphs, and the equations. 7. At which months does Theresa have more money? 8. At which months does Cary have more money? 9. At which month do they have the same amount of money? What do you notice about the graphs at this month? LF6 SP3
6.1 Introduction to Systems of Linear Equations SYSTEMS OF EQUATIONS A system of equations (or simultaneous equations) is a set of two or more equations in the same variables considered at the same time. In this unit we will learn about systems of two linear equations in two variables. To solve a system of equations, find the values of the variables that satisfy both equations simultaneously. A system of equations is consistent if it has at least one solution. A system of two linear equations in two variables is consistent if it has (a) exactly one solution (independent) or (b) infinitely many solutions (dependent). A system of two equations is inconsistent if it has no solutions Two systems of linear equations are graphed below. Refer to the graphs and write whether each system of equations appears to have exactly one solution (independent) or no solutions (inconsistent). Explain your reasoning for each answer. 1. 3. Sketch the graphs for the following system of linear equations. Hint: change the second equation to slope-intercept form before graphing. y = 2x + 3 2y = 4x + 6 Why do you think we say that this system has infinitely many solutions (dependent)? 2. LF6 SP4
6.1 Introduction to Systems of Linear Equations SOLVING SYSTEMS OF EQUATIONS BY GRAPHING For each system, change equations to slope-intercept form when needed, graph them, and then describe the solutions. 1. 3y = 9x + 12 y = 3x 5 2. y 8 = 2 5 x 2x + y = -4 LF6 SP5
6.1 Introduction to Systems of Linear Equations SOLVING SYSTEMS OF EQUATIONS BY GRAPHING (continued) For each system, change equations to slope-intercept form when needed, graph them, and then describe the solutions. 3. y = -3x + 5 y = 1 x 2 4. y 4x = 12 2x 1 2 y = 6 5. How many solutions will each of the follwing systems have descibed below? Make conjectures and justify through discussion with your peers. Diagrams may be helpful. a. Two lines with the same slope and different y-intercepts. b. Two lines with the same slope and the same y-intercepts. c. Two lines with different slopes and the same y-intercepts. LF6 SP6
6.2 Solving Systems Using Algebra Ready (Summary) SOLVING SYSTEMS USING ALGEBRA We will solve systems of linear equations by the substitution method and the elimination method. Go (Warmup) Set (Goals) Solve systems of linear equations using tables, graphs, and algebraic notation. Compare the effectiveness of different strategies for solving systems of equations. Understand and be able to use the substitution and elimination methods for solving systems. Use the addition property of equality, the multiplication property of equality, and the substitution property to manipulate expressions and equations The transitive property of equality states that if a = b and b = c, then a = c. 1. Given: Andre is the same height as Betty. Betty is the same height as Cleo. What can we conclude about the relative heights of Andre and Cleo? A variation of the transitive property of equality is the substitution property of equality, which states that if a = b, then a can replace b in any equation or inequality. 2. Given: = 2 (one heart is equal to two clouds) For the equation below, substitute so that the right side of the equation has only clouds and numbers, no hearts. 3 + 5 = 3( ) + 5 Simplify the right side of the equation: LF6 SP7
6.2 Solving Systems Using Algebra GOING TO THE PARK REVISITED Two friends, Malcolm and Zeke, are going to the park that is near their school. They leave at the same time from different locations. Malcolm is 30 meters from the school on his way to the park, and he is walking at the rate of 1 meter per second. Zeke is at school and will jog to the park at a rate of 4 meters per second. 1. Fill in the table and graph the data. Use one color for Malcolm s graph and another for Zeke s. Label and scale the axes appropriately. Time in seconds (x) Malcolm s distance from school in meters (y) Time in seconds (x) 0 30 0 1 1 2 2 3 3 x x Zeke s distance from school in meters (y) 2. Based upon the table, at what number of seconds are the two boys the same distance from school? How do you know? 3. Based upon the graphs, at what number of seconds are the two boys the same distance from school? How do you know? LF6 SP8
6.2 Solving Systems Using Algebra GOING TO THE PARK REVISITED (continued) 4. Write equations for the graphs of both boys travels (distance in terms of time). Malcolm: Zeke: y = y = 5. We are focusing on the number of seconds at which the two boys are the same distance from school. From the information in problem 4, why is it true that x + 30 = 4x? 6. Solve for x. 7. What does the solution for x mean in the context of the problem? 8. From above, since x =, use the substitution property to solve for y in the equations for Malcolm and Zeke and simplify: Malcolm: Zeke: y = = y = = 9. What do the solutions for y mean in these equations? 10. After how many seconds are the boys the same distance from school? What is this distance? 11. Looking at the algebraic substitution process above, does the number of seconds at which the boys are the same distance from school match what is in the table and the graphs above? LF6 SP9
6.2 Solving Systems Using Algebra SOLVING SYSTEMS USING SUBSTITUTION 1. Solve the following system of equations by following the instructions and answering the questions below. y = -4x + 2 y = 8x 1 a. Graph the system. b. Ike looked at the graph and said, I can t tell for sure what the solution is. Explain why you think Ike said this. c. What do you think the solution is? d. Since the expressions -4x + 2 and 8x 1 are both equal to y, use the transitive property to write a new equation that is in one variable, namely x. Then solve for x. = e. From above, since x =, use the substitution property to solve for y in both of the original equations. y = -4x + 2 y = 8x 1 y = -4( ) + 2 y = 8( ) 1 f. Does y have the same value for both equations in problem 4? g. What is the solution of the system? (, ) LF6 SP10
6.2 Solving Systems Using Algebra SOLVING SYSTEMS BY SUBSTITUTION (continued) 2. Solve the system below by following the instructions and answering the questions below. y = 2x 1 2y 3x = -5 a. Circle an expression from the first equation that is equal to y. b. Use the substitution property and solve for x. 2y 3x = -5 2( ) 3x = -5 c. From above, since x =, use the substitution property to solve for y in both of the original equations. y = 2x 1 2y 3x = -5 d. What is the solution of the system? (, ) 3. Solve the following system using the substitution method. Note that there is not a variable isolated ( by itself ) in either equation. 2x = -2 + 2y 3x 4y = -2 LF6 SP11
6.2 Solving Systems Using Algebra SOLVING SYSTEMS BY SUBSTITUTION PRACTICE Solve each system using the substitution method. 1. y = 3x + 1 y = x + 5 3. 2x 3y = 15 x + y = 6 2. y 2x = 6 y = 2x + 2 4. 4x = 6 + y 2x 1 2 y = 3 LF6 SP12
6.2 Solving Systems Using Algebra USING PROPERTIES OF EQUALITY TO MANIPULATE EQUATIONS For all numbers a, b, and c, if a = b, then a + c = b + c addition property of equality (equals added to equals are equal) AND Given m = n. Write true (T) or false (F) next to each statement. ac = bc multiplication property of equality (equals multiplied by equals are equal) 1. m + 3 = n + 3 2. m + (2 + 5) = n + 7 3. 6m = 6n 4. m 4 = n 4 5. m (2 6) = n 4 6. 7. m n = -6-6 m 3 = 3 n 8. 2(m + 5) = 2n + 10 9. m 8 = 8 n For each problem, given two equations, write a new equation. (Note that equations are written in a vertical orientation to make adding and subtracting expressions easier.) 10. (3 + 6) = (9) 11. +(2 + 5) = +(7) + = Why can you add 2 + 5 to the left side of the first equation and 7 to the right side? (3 + 6) = ( 9) -(2 + 5) = ( 7) + = Why can you subtract 2 + 5 from the left side of the first equation and 7 from the right side? LF6 SP13
6.2 Solving Systems Using Algebra Use different colored pencils for each graph. 1. Graph this system of equations: 2x+ y = 5 x+ 2y = 4 2. What coordinates represent the solution to this system? (, ) 3. Solve the system above using the substitution method. SOLVING SYSTEMS REVIEW As an ordered pair, the solution to the system is: (, ) As equations, the solution to the system is: x = y = 4. Consider the solutions in the box above as equations of lines. Graph these two equations on the same grid above. What do you notice? LF6 SP14
6.2 Solving Systems Using Algebra CREATING EQUATIONS USING PROPERTIES OF EQUALITY On the previous page, you solved the system 2x+ y = 5 x+ 2y = 4 Use different colored pencils for each graph. 1a. Apply the addition property of equality to this system to arrive at another equation. Graph this equation on the previous page. Slope-intercept form may be helpful. 1b. What do you notice about the graph of the new equation? 2a. Multiply both sides of the second equation by -2 to arrive at another equation. Graph this equation on the previous page. Slope-intercept form may be helpful. 2b. What do you notice about the graph of the new equation? 3a. Add the expressions for the equation created in problem 1a to the expressions for the equation created in problem 2a to arrive at another equation. Graph this equation on the previous page. Slope intercept form may be helpful. 3b. What do you notice about the graph of the new equation? 4a. Use properties of equality to create another equation from the original ones. Then graph this new equation. Slope intercept form may be helpful. 4b. What do you notice about the graph of this new equation? in two different ways. (2x + y) = (5) +( x + 2y) = +(4) = from 1a: -2(x + 2y) = -2(4) = from 2a: LF6 SP15
6.2 Solving Systems Using Algebra SOLVING SYSTEMS BY ELIMINATION Given a system of two linear equations, properties of equality can be used to create new systems that have the same solution as the given ones. Using this idea strategically allows for manipulating the equations to eliminate one of the variables. This process is called solving a system by elimination. Consider the following system: 2x+ y = 10 (1) y = x+ 2 (2) 1. Here is one strategy for using elimination to solve this system. Directions Rewrite both equations in the form ax + by = c, changing equations (1) and (2) to (3) and (4). Apply the addition property of equality to equations (3) and (4) to eliminate y. This is now equation (5) Solve for x. Use substitution in equation (1) to solve for y. Write the solution as equations and as an ordered pair. Check the solution in equation (2). (1 3) (2 4) (5) Work LF6 SP16
6.2 Solving Systems Using Algebra SOLVING SYSTEMS BY ELIMINATION (continued) Consider the same system: 2x+ y = 10 (1) y = x+ 2 (2) 2. Here is another strategy for using elimination to solve this system. Rewrite both equations in the form ax + by = c. (1 3) (2 4) Apply the multiplication property of equality by multiplying both sides of equation (4) by -2. (5) Apply the addition property of equality to equations (3) and (5) to eliminate x. Solve for y. Use substitution into equation (1) to find x. Write the solution as equations and as an ordered pair. Check the solution in equation (2). LF6 SP17
6.2 Solving Systems Using Algebra SOLVING SYSTEMS BY ELIMINATION (continued) Solve the following system of equations. 3. Given system: y+ x = 5 y = 6 2x a. Rewrite both equations in the form ax + by = c b. Solve by eliminating the y s c. Solve by eliminating the x s d. Write the solution as equations and as an ordered pair. 4. Given system: 4x 3y = 1 3y = 2 2x a. Rewrite both equations in the form ax + by = c b. Solve by eliminating the y s c. solve by eliminating the x s d. Write the solution as equations and as an ordered pair. LF6 SP18
6.2 Solving Systems Using Algebra SOLVING SYSTEMS BY ELIMINATION PRACTICE Solve each system using the elimination method. Check by substitution or by solving again for the other variable. 1. 3x + y = 12 y 2 = -5x 3. 4x 3y = 12 3y = 4x 6 2. 3x = y + 3 2y = 5x 1 4. 2y = 5x 1 2 10x = -4y LF6 SP19
6.2 Solving Systems Using Algebra SOLVING WORD PROBLEMS Read each problem. Define the variables. Translate each word problem into a system of equations. Solve the system by using the substitution or elimination method. Clearly state the solutions. Check solutions in the original problem. 1. Sid and Nancy went to the same store. Sid bought 4 shirts and 3 pairs of jeans for $181. Nancy bought 1 shirt and 2 pairs of jeans for $94. All shirts were the same price and all jeans were the same price. How much did shirts and jeans each cost? 2. The sum of two numbers is 79. Three times the first number added to 5 times the second number is 283. What are the two numbers? LF6 SP20
6.3 Solving Systems Linear Inequalities SOLVING SYSTEMS OF LINEAR INEQUALITIES Ready (Summary) We will graph linear inequalities, and then solve systems of linear inequalities by graphing. Then we will solve a problem using the fourfold way (numbers, pictures, symbols, and words). Go (Warmup) Set (Goals) Graph each inequality. Be sure they are in slope-intercept form first. 1. y > - 1 2 x + 3 2. 3x y 1 Solve systems of linear inequalities by graphing Use the fourfold way to show complete solutions to a complex problem 3. What is important to remember to do when graphing an inequality with the sign versus one with the < sign? 4. What is important to remember to do when multiplying or dividing both sides of an inequality by a negative value? LF6 SP21
6.3 Solving Systems Linear Inequalities SOLVING SYSTEMS OF INEQUALITIES For each system, graph both inequalities. Test several points to verify correct shading. 1. y -x + 1 y > 1 3 x 2. y 2 < -4x -3x + y 6 LF6 SP22
6.3 Solving Systems Linear Inequalities SOLVING SYSTEMS OF INEQUALITIES (continued) For each system, graph both inequalities. Test several points to verify correct shading. 3. 2y x 6 4 + x 2y 4. x < y 2 y < 3x LF6 SP23
6.3 Solving Systems Linear Inequalities PRACTICE SOLVING SYSTEMS OF INEQUALITIES For each system, graph both inequalities. Test several points to verify correct shading. 1. y -3x + 5 y > 1 x 2 2. y 8 < 2 5 x 2x + y -4 LF6 SP24
6.4 Vocabulary, Skill Builders, and Review Here are four linear functions: FOCUS ON VOCABULARY (a) y = 2x + 3 (b) y = 2x + 6 (c) y = -2x + 6 (d) 2y = 12 4x Work space: Complete each statement using mathematical vocabulary from the word bank and this packet. 1. Equation (a) and (b) have the same. 2. Equation (b) and (c) have the same. 3. (0, 6) is a to the system of equations (b) and (c). 4. Equations (a) and (b) are a(n) system of equations because they have no solution. 5. Equations (b) and (c) are a(n) system of equations because they have a exactly one solution 6. Equations (c) and (d) are a(n) system of equations because they have infinite solutions. LF6 SP25
6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 1 Write each equation in slope intercept form. If the equation is already in slope-intercept form, then circle the equation. 1. 4x 7 = y 3. 2. -y = 1 2 x + 5 1 2 y = 3x 1 4. -x + 1 4 y = 2 LF6 SP26
6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 2 For each problem, translate it into an inequality and solve it. Then check the boundary point and another value. 1. Twice a number is less than the number. Find the possibilities for the number. Write inequality and solve: Answer in words: Check: 2. The product of a number and -2 is greater than or equal to 3 less than the number. Find the possibilities for the number. Write inequality and solve: Answer in words: Check: Lauren has 35 coins. Some are quarters and some are dimes. The total value of the coins is $6.65. How many quarters and how many dimes does she have? LF6 SP27
6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 Solve the following systems of equations by graphing. Clearly list and describe the solutions, if any. 1. y = - 2x + 2-3y 9 = x 2. 1 2 x + y = 3 2y = 6 x LF6 SP28
6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 4 Write each statement using symbols. If the variable is on the right side, change it to the left side using appropriate properties. Then graph each. 1. x is equal to -3 2. x is greater than -1 3. the opposite of x is less than or equal to 1 4. 0 is greater than x 5. -2 is less than or equal to the opposite of x. Graph each inequality. Be sure they are in slope-intercept form first. 6. y 3x 2 7. y 2x < 1 8. Describe the differences between the graph of an inequality in one variable with the graph of an inequality in two variables. LF6 SP29
6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 5 Here is information about some linear functions. Use tables, graphs, and/or equations to answer the questions. Line A Goes through (3, 3) and (2, 0) Line B Goes through (5, -1) and (3, -2) Line C Table of Values x y 0 3 1 6 2 9 3 12 4 15 5 18 6 21 1. Do lines A and B intersect? If yes, where? If no, explain why. 2. Do lines C and D intersect? If yes, where? If no, explain why. 3. If you have not already done so, write equations for lines A, B, and C above. Make a table of values for line D above. Graph all lines to the right. Recall that a direct proportion is a relationship where one variable is a multiple of the other (also called a direct variation). 4. Which line appears to represent a direct variation? What appears to be the multiple? Line D y = 3x 5. Which line goes through the origin? LF6 SP30
6.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Solve each system using the substitution method or the elimination method. 9. y + 3x = 1 2x 4 = y 11. x + y = -8 y x = -4 10. 1 x 2y = -3 3 y = 3 12. y = -6x + 10 3x + 1 2 y = -1 LF6 SP31
6.4 Vocabulary, Skill Builders, and Review TEST PREPARATION Show your work on a separate sheet of paper and choose the best answer. 1. What is the solution to this system of equations? 2x y = 1 4 1 x + 4y = 3 2 A. 3 1, 4 2 B. 1 3, 2 4 C. 3 1, 4 2 2. A system of linear equations that has exactly one solution is E. consistent, dependent F. consistent, independent G. inconsistent, dependent D. 3 1, 4 2 H. inconsistent, independent 3. An example of a graph of a system of linear equations that has infinitely many solutions is A. A pair of lines intersecting in exactly one point B. A pair of parallel lines C. A pair of coinciding lines 4. Consider this system. Just by looking at it, how do y = -2x 1 you know that it has no solutions? y = -2x + 6 E. Both equations are linear F. Both equations are in slopeintercept form G. Both equations have the same slope and different y- intercept D. Three lines intersecting in exactly one point H. It is a consistent, dependent system LF6 SP32
6.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK Show your work on a separate sheet of paper and write your answers on this page. 6.1 Introduction to Systems of Linear Equations 1. Solve this system by graphing. y + x = -2 y = -x 2 6.2 Solving Systems Using Algebra 2. Solve this system by using the substitution method. -4 = 2x y -9 = -3x + y 3. Solve this system by using the elimination method. -9 = 4x + y -2y = 3x + 10 6.3 Solving Systems of Linear Inequalities 4. Solve this system by graphing. y < -2x 1 2x + y 4 LF6 SP33
This page is left intentionally blank for notes. LF6 SP34
This page is left intentionally blank for notes. LF6 SP35
This page is left intentionally blank for notes. LF6 SP36
HOME-SCHOOL CONNECTION Here are some questions to review with your young mathematician. 1. Solve this system of equations using substitution or elimination. Describe why you chose your method. 2. Solve this system of inequalities by graphing. Describe the difference between a solid line and a dashed line. Explain what the double-shaded region means. Parent (or Guardian) Signature 2y = -x + 4 3y + x = -2 y < -x + 1 y 2x + 1 LF6 SP37
COMMON CORE STATE STANDARDS MATHEMATICS 8.EE.8a 8.EE.8b 8.EE.8c A-CED-2* A-REI-1 A-REI-5 A-REI-6 A-REI-10 A-REI-12 MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8 Analyze and solve pairs of simultaneous linear equations: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Analyze and solve pairs of simultaneous linear equations: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. Analyze and solve pairs of simultaneous linear equations: Solve real-world and mathematical problems leading to two linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. SELECTED COMMON CORE MATHEMATICS STANDARDS Look for and make use of structure. Look for and express regularity in repeated reasoning. LF6 SP38