Solving Systems of Linear Inequalities Focus on Modeling

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Name Class 5-6 Date Solving Systems of Linear Inequalities Focus on Modeling Essential question: How can you use systems of

4.1 Y ou are purchasing jeans and T-shirts. Jeans cost $35 and T-shirts cost $15.

1 Write a system of linear equations to model the situation.

B Write an expression to represent the amount you will pay for y T-shirts at $15 per shirt.

Use this verbal model and your expressions from Steps 1A and 1B to write an equation for the total amount you will spend.

1 Name Class 5-6 Date Solving Systems of Linear Inequalities Focus on Modeling Essential question: How can you use systems of linear equations or inequalities to model and solve contextual problems? N-Q.1.1*, N-Q.1.*, A-CED.1.*, A-CED.1.3*, A-REI.3.6, A-REI.4.1 Y ou are purchasing jeans and T-shirts. Jeans cost $35 and T-shirts cost $15. You plan on spending $115 and purchasing a total of 5 items. How many pairs of jeans and how many T-shirts can you buy? 1 Write a system of linear equations to model the situation. A Write an expression to represent the amount you will pay for x pairs of jeans at $35 per pair. B Write an expression to represent the amount you will pay for y T-shirts at $15 per shirt. C The total amount spent for jeans and T-shirts is given below in words. Use this verbal model and your expressions from Steps 1A and 1B to write an equation for the total amount you will spend. Amount Spent for Jeans + Amount Spent for T-shirts + = Total Amount Spent = D What variable represents the number of pairs of jeans purchased? E What variable represents the number of T-shirts purchased? F Write an equation to represent the total number of items purchased. G Write a system of linear equations to model the situation. Chapter Lesson 6

2 REFLECT 1a. What units are associated with the expressions that you wrote in 1A and 1B? 1b. When you add the units for the expressions representing the amounts spent on jeans and T-shirts, what units do you get for total amount spent? Are they the units you expect? Solve the system algebraically. A B Solve an equation for one variable. x + y = 5 y = Select one of the equations. Isolate the variable y on one side. Substitute the expression for y into the other equation and solve. 35x + 15 ( ) = 115 Substitute the expression for the variable y. 35x = 115 Use the Distributive Property = 115 Combine like terms. = 40 Subtract from each side. C x = Divide each side by. Substitute the value of the variable you found in Part B into one of the equations and solve for the other variable. + y = 5 Substitute the value you found into an equation. y = Subtract from each side. So, REFLECT is the solution of the system. a. In the solution, what does the x-value of the ordered pair represent in the context of the situation? What does the y-value represent? b. Explain why substitution is a good method to use to solve this system. Chapter Lesson 6

3 3 Check the solution by graphing. A Graph each equation. Step 1: Find the intercepts for 35x + 15y = 115 and graph the line. x-intercept: y-intercept: Step : Find the intercepts for x + y = 5 and graph the line. x-intercept: y-intercept: T-shirts y x B Find the point of intersection. The two lines appear to intersect at Pairs of jeans 8 REFLECT 3a. What units are represented on the x-axis? 3b. What units are represented on the y-axis? 3c. Does the solution you found by graphing confirm that the solution you found algebraically was correct? Explain. 3d. Was it easier to solve the system algebraically or by graphing? Explain your reasoning. 4 Interpret the solution. A B C What does the solution tell you about the number of pairs of jeans and the number of T-shirts you can purchase? In the context of the problem, what could be the values of x and y? Is the solution reasonable? Explain your reasoning. Chapter Lesson 6

4 REFLECT 4a. Is the solution you found the only solution for this linear system? Explain how you know. EXTEND 1. Suppose you want to buy at least 5 items and spend no more than $115. How can you modify the system of linear equations you wrote to model this new situation?. Write an inequality to represent buying at least 5 items. 3. Write an inequality to represent spending no more than $ Are there any other conditions on the system, based on the context of the problem? If so, what are they? 5. Write a system of linear inequalities to model the situation. Include any new conditions from Question What constraints do the additional conditions based on the context of the problem place on where in the plane the solution region will be located? Chapter Lesson 6

5 7. Graph the system of inequalities. Step 1 Graph x + y 5. The equation of the boundary line is. x-intercept: y-intercept: The inequality symbol is, use a line. T-shirts y Shade the boundary line, because (0, 0) is not a solution of the inequality x 8 Step Graph 35x + 15y 115. Pairs of jeans The equation of the boundary line is. x-intercept: y-intercept: The inequality symbol is, use a line. Shade the boundary line line, because (0, 0) is a solution of the inequality. Step 3 Identify the solutions. The solutions of the system are represented by the that form a to the of the y-axis. shaded regions 8. In the context of the situation, are all points in the overlapping shaded region possible solutions? Why or why not? Explain. 9. Is the ordered pair that was the solution of the system of linear equations for this situation a solution of this system of inequalities? 10. If you buy at least 5 items and spend no more than $115, what is the greatest number of jeans you can buy? Explain your reasoning. 11. If you buy at least 5 items and spend no more than $115, what is the greatest number of T-shirts you can buy? Explain your reasoning. Chapter Lesson 6

6 1. Use the graph to make a list of all the possible solutions for the number of pairs of jeans and number of T-shirts you can purchase if you buy at least 5 items and spend no more than $115. Pairs of Jeans T-Shirts Total Items Total Cost Chapter Lesson 6

Name Class Date Additional Practice 1. (, ); < 3. (, 5); > + 1 > 3. + (1, 3); + > 4 1 5-6 4. + 4 5. 1 + 1 + < 3 6. > < + 4 a. a. a. b. b. b. 7.

7 Name Class Date Additional Practice 1. (, ); < 3. (, 5); > + 1 > 3. + (1, 3); + > < 3 6. > < + 4 a. a. a. b. b. b. 7. Charlene makes $10 per hour babysitting and $5 per hour gardening. She wants to make at least $80 a week, but can work no more than 1 hours a week. a. Write a system of linear equations. b. Graph the solutions of the system. c. Describe all the possible combinations of hours that Charlene could work at each job. d. List two possible combinations. Chapter Lesson 6

Problem Solving 1. Paul earns $7 per hour at the bagel shop and $1 per hour mowing lawns. Paul needs to earn at least $10 per week, but he must work less than 30 hours per week.

8 Problem Solving 1. Paul earns $7 per hour at the bagel shop and $1 per hour mowing lawns. Paul needs to earn at least $10 per week, but he must work less than 30 hours per week. Write and graph the system of linear inequalities that describes this situation.. Zoe plans to knit a scarf. She wants the scarf to be more than 1 but less than 1.5 feet wide, and more than 6 but less than 8 feet long. Graph all possible dimensions of Zoe s scarf. List two possible combinations. 3. Which system of linear inequalities represents the graph? A 4 C B 4 D If 6 buffet tables are built, which can NOT be the number of dining tables built? F 4 H 8 G 6 J 10 Chapter 5 31 Lesson 6

Algebra 1 Semester Exam

Algebra 1 Semester Exam 015-016 30 questions Topics: Expressions, Equations, and Inequalities Functions inear Function Systems of Equations and Inequalities Reporting Category 1: (50% of the Semester Exam)