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STUDY GUIDE REVIEW MODULE

Solving Systems of Linear Equations 11

Essential Question: How can you use a system of linear equations to solve real-world problems?

KEY EXAMPLE (Lesson 11.2)

Solve ⎧ ⎨ ⎩ 4x + y = 7

-6x + y = -3 by substitution.

Solve an equation for one variable.

-6x + y = -3 Select one of the equations.

y = 6x -3 Solve for y. Isolate y on one side.

Substitute the expression for y in the other equation and solve.

4x + (6x -3) = 7 Substitute the expression for y.

10x -3 = 7 Combine like terms.

10x = 10 Add 3 to both sides.

x = 1 Divide each side by 10.

Substitute the value for x into one of the equations and solve for y.

4(1) + y = 7 Substitute the value of x into the first equation.

4 + y = 7 Simplify.

y = 3 Subtract 4 from both sides.

So, (1, 3) is the solution of the system.

KEY EXAMPLE (Lesson 11.3)

Solve ⎧ ⎨ ⎩ -6x + 8y = 19

6x - 8y = -19

by adding.

Add the equations.

-6x + 8y = 19

+6x - 8y = -19―― 0 + 0 = 0

0 = 0

The resulting equation is always true, so the system has infinitely many solutions.

Key Vocabularyelimination method

(eliminación)substitution method

(sustitución)system of linear equations

(sistema de ecuaciones lineales)

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Mathematical Practices: MP.1, MP.2, MP.3, MP.4, MP.6A-REI.6, A-CED.3

Study Guide Review

SCAFFOLDING SUPPORT

total cost = purchase price + gas used (gal) ____________

year ∙ cost ($)______gal ∙ years driven

ASSESSMENT AND INTERVENTION

MODULE PERFORMANCE TASK

SUPPORTING STUDENT REASONING

The prices of the cars:

The distance (in miles) the cars are driven each year:

The miles per gallon each car gets:

The cost of gas:

527 Module 11

M O D U L E 11

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est mpg25/35

msrp$22,600

est mpg43/39

msrp$25,990

LE Hybrid LE

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MODULE PERFORMANCE TASK

Do Hybrid Cars Pay for Themselves?Your family wants to buy a specific model of new car and is considering buying the hybrid version. Use the information shown for the two cars to determine how long it will take to save enough money on gas to pay for the extra cost of the hybrid. Then make a recommendation on which car to buy.

Start by listing in the space below the information you will need to solve the problem. Then use your own paper to complete the task. Be sure to write down all your data and assumptions. Then use graphs, numbers, words, or algebra to explain how you reached your conclusion.

EXERCISESSolve each system of equations. (Lessons 11.1, 11.2, 11.3, 11.4)

1. ⎧ ⎨

⎩ 3x + 7y = -5

8x + 9y = 6

2. ⎧ ⎨

⎩ -5x + 2y = 13

3x - 2y = -11

3. ⎧ ⎨

⎩ 9x - 2y = -5

-6x + y = -1 4.

⎧ ⎨ ⎩ 7x - 9y = -11

7x - y = -9

5. ⎧ ⎨ ⎩ -3x + 5y = 8

3x - 5y = -8

6. ⎧ ⎨ ⎩ -2x + 6y = 6

-4x - 8y = 12

(-3, 0)infinitely many solutions

(- 5 __ 4 , 1 __ 4 ) ( 7 __ 3 , 13)

(-1, 4)(3, -2)

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(7.0, 35.2)

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)Time (years)

y

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40

50

20

30

10

102 4 6 8

x

DISCUSSION OPPORTUNITIES

SAMPLE SOLUTIONAssumptions

Purchase price: $22,600 (non-hybrid) and $25,990 (hybrid)

Annual miles driven: 13,476

Average mpg: 30 mpg (non-hybrid) and 41 mpg (hybrid)

Cost of gas: $4 per gallon

Equations

Let x = number of years and y = total cost.

Non-hybrid:

y = 22,600 + ( 13,476 _______ 30 ) (4) x

Hybrid:

y = 25,990 + ( 13,476 _______ 41 ) (4) x

The y-intercepts represent the purchase prices of the two cars, and the intersection, about (7.0, 35.2), indicates that it will take about 7 years to save enough on gas to pay for the extra cost of the hybrid.

Recommendations may vary. See notes under “Discussion Opportunities.”

Assessment Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain.0 points: Student does not demonstrate understanding of the problem.

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