LESSON 17 An Introduction to Sstems of Equations LEARNING OBJECTIVES Toda I am: completing the Desmos activit Sstems of Two Linear Equations. So that I can: write and solve a sstem of two linear equations to explore the numerical and graphical meaning of solution. I ll know I have it when I can: write an equation of a line to describe Duke and Shirle s walk on a ramp. Opening Exercise 1. Go to student.desmos.com and tpe in our teacher s code: You ll be completing the activit, Sstems of Two Linear Equations. Complete the questions below as ou go through the activit. 2. On Screen 18, ou wrote two equations where the lines do not intersect. What is true about these two lines? 3. Write a new sstems of equations where the lines will not meet. alekleks/shutterstock.com 4. In general, what can we sa about lines that don t meet? What must be true about them? 437
438 Module 2 Solving Equations and Sstems of Equations. For each set of lines below, determine which ones will have a solution and which will not. A. x 18 B. x 3 C. 2x 2 8 D. 3x 3 18 x 0 x 0 x 0 x 0 Mental Math In man of these problems, the solution can be determined b thinking about the relationship between the s. Often ou can use mental math to determine the solution. Tr it out in Exercises 6 and 7. 6. Write a sstem of equations where the sum of two s is and the difference is 6. Can ou determine the solution without graphing? Explain our reasoning. 7. Write a sstem of equations where the sum of the two s is and the difference is 3. Can ou determine the solution without graphing? Explain our reasoning. A nnmmbbefrt xt.to 87 8 8,2 I 6 f 2 FIJI'S 6,3 Continuous and Discrete Data Points Discrete data points can be numeric like the of dogs but it can also be categorical like colors or gender. Continuous data are not restricted to defined separate values, but can take on ANY value over a continuous range. Think about which tpe of data points are given in Exercises 8 and 9. 8. Richard thinks of two s. The onl clue he gives is The sum of two s is. What are the s Richard could be thinking of? Continuous X A. Create an equation using two variables to represent this situation. Be sure to explain the meaning of each variable. l 2 Xt lo B. List at least six solutions to the equation ou created in Part A. I 8,2 C. Create a graph that represents the solution set to the equation.
9. Gia had songs in a plalist composed of songs from her two favorite artists, Beoncé and Jennifer Lopez. How man songs did she have b each one in the plalist? Unit 4 Sstems of Equations and Inequalities 439 Lesson 17 An Introduction to Sstems of Equations A. Create an equation using two variables to represent this situation. Be sure to explain the meaning of each variable. JStone/Shutterstock.com; Andrea Raffin/Shutterstock.com boesoonndes Itb O b Discrete B. List at least three solutions to the equation ou created in Part A. b TSB 4613 9 JIB 7313 C. Create a graph that represents the solution set to the equation.. Compare our solutions to Exercises 8 and 9. How are the alike? How are the different? Exercise 8 Exercise 9 finite set As the stand, Exercises 8 and 9 are not sstems of equations, since there is onl one equation for each. 11. For Exercise 8, graph an additional clue from Richard, The difference of m two s is 6. Use the graph to determine Richard s two s. X 6 6 8.2,4 12. Gia sas that the difference in the of songs is 6. Use the graph in Exercise 9 to find the of Beoncé songs and the of Jennifer Lopez songs Gia has on her plalist. j infinite amt ofpossibilities decimals 8 2 solution 8,2 7,1 9,3 8 2
440 Module 2 Solving Equations and Sstems of Equations Throughout this unit, ou ll be writing sstems of equations or inequalities with different real-world problems. s look at a stor of two people walking along a ramp. Duke starts at the base of a ramp and walks up it at a constant rate. His elevation increases b 3 ft. ever second. Just as Duke starts walking up the ramp, Shirle starts at the top of the same 2 ft. high ramp and begins walking down the ramp at a constant rate. Her elevation decreases 2 ft. ever second. Q Wang/Shutterstock.com Understanding the Problem 13. Fill in the chart below to help ou understand how quickl Duke and Shirle move and where the start. Duke s Path Shirle s Path Time in Seconds Elevation in Feet Time in Seconds Elevation in Feet 0 0 0 2 1 3 1 23 2 2 3 3 4 4 6 6 7 7 8 8 9 11 12 Elevation in Feet 2 24 23 22 21 20 19 18 17 16 1 14 13 12 11 9 8 7 6 4 3 2 1 0 The Stor of Duke and Shirle 0 1 2 3 4 6 7 8 9 11 12 13 14 1 Time in Seconds
Unit 4 Sstems of Equations and Inequalities 441 Lesson 17 An Introduction to Sstems of Equations 14. Use the extra row in each table to write an expression that describes the elevation pattern. 1. Use the grid to graph the data for Duke and Shirle. Be sure to add a legend to help the reader understand our graph. Writing an Equation of a Line The equation of a line can be in the form mx b, where m represents the slope of the line and b represents the -intercept. 16. Determine the slope of Duke s line and Shirle s line. Wh is one of the slopes negative? Duke s slope: Shirle s slope: 17. What is the -intercept for each line? Duke s -intercept: Shirle s -intercept: 18. Write the equation of Duke s line and Shirle s line. Duke s line: Shirle s line: 19. Where do the two lines intersect? This is the point of intersection. What time do Duke and Shirle pass each other? 20. How could ou use the equations to find the point of intersection? The equations ou wrote in Exercise 18 are in slope-intercept form or mx b. Joint Slope Point-slope is another ver useful form of a linear equation. For point-slope we need an point on the line, not just the -intercept, and the slope of the line. The general form looks like 1 m(x x 1 ) where (x 1, 1 ) is an point on the line and m is the slope of the line. write equation 21. A. Use the points when the time is 3 seconds and the slopes from Exercise 16 with the pointslope equation to get equations for Duke and Shirle. B. How do these equations compare to the ones ou wrote in Exercise 18?
442 Module 2 Solving Equations and Sstems of Equations Lesson Summar A sstem of equations is a set of equations that describe a situation. Example: Two s have a sum of 12 and a difference of 4. What are those two s? Define our variables. Use the information in the problem to write two equations. x the first the second x 12 x 4 In later lessons, ou ll solve sstems of equations using graphing, substitution and a new method, elimination. Linear Equations: If ou know the slope and -intercept, ou can use the equation mx b to write an equation for the line. If ou have the slope and an point on line ou can use the point-slope equation, 1 m(x x 1 ) where (x 1, 1 ) is an point on the line and m is the slope of the line. Example 1: The slope of a line is 3 and the -intercept is. An equation of this line is. Example 2: The slope of a line is ½ and the point (2, 4) is on the line. The equation of this line is. Point of Intersection: The point of intersection is an ordered pair that is a solution to both equations. On a graph, this is where the two graphs cross each other. Example: Determine the point of intersection for the graph at the right. Point of intersection: 37 4 126 2 2,6
Unit 4 Sstems of Equations and Inequalities 443 Lesson 17 An Introduction to Sstems of Equations NAME: PERIOD: DATE: Homework Problem Set Write a sstem of equations for Problems 1 and 2. Be sure to define our variables. Then create a table and a graph of each problem and find a solution. 1. The difference of two s is 3 and their sum is 13. What are the two s? 2. The difference of two s is and the sum is 4. What are the two s? Sstem of Equations: Sstem of Equations: 1st 2nd Sum of s 1st 2nd Difference of s 1st 2nd Sum of s 1st 2nd Difference of s 13 3 4 13 3 4 13 3 4 13 3 4 Solution: Solution:
444 Module 2 Solving Equations and Sstems of Equations Write a sstem of equations for Problems 3 and 4. Be sure to define our variables. You do NOT need to solve these problems. 3. Jack and his sister, Malonie, are 4 ears apart in age. The sum of their ages is 28. What are their ages? 4. Two of Julie s textbooks are a total of $6. The difference in price between the two books is $9. What is the cost of each book? Sstem of Equations: Sstem of Equations: pixelheadphoto digitalskillet/shutterstock.com Vetal/Shutterstock.com
Unit 4 Sstems of Equations and Inequalities 44 Lesson 17 An Introduction to Sstems of Equations Spiral REVIEW Determining Slope and -intercept from Graphs Find the slope and -intercept of each line.. slope -intercept 6. slope -intercept 7. slope -intercept 8. slope -intercept 9. slope -intercept. slope -intercept 11. slope -intercept 12. slope -intercept 13. slope -intercept
446 Module 2 Solving Equations and Sstems of Equations Spiral REVIEW Solving Equations Solve each equation. 14. 16 6 2(8 7) 1. 2(6b 8) 4 6b Spiral REVIEW Graphing Lines 16. Match each equation with its graph. Explain our reasoning. A. x 6 B. x 2 12 C. 2x 4 D. 3x 6 E. x 4 1 2 - - x - - x - - - - 3 4 - - x - - x - - x - - - - - -