6 Pull It All Together

BIG idea Solving Equations and Inequalities

There are several ways to solve systems of equations and inequalities, including graphing and using equivalent forms of equations and inequalities within the system. The number of solutions depends on the type of system.

Task 1

Solve using two different methods. Explain which method you found to be more efficient.

1. 3 x − 9 y = 3 6 x − 3 y = − 24 table with 2 rows and 1 column , row1 column 1 , 3 x minus 9 y equals 3 , row2 column 1 , 6 x minus 3 y equals negative 24 , end table
2. 7 x − 3 y = 20 5 x + 3 y = 16 table with 2 rows and 1 column , row1 column 1 , 7 x minus 3 y equals 20 , row2 column 1 , 5 x plus 3 y equals 16 , end table
3. y = 1 2 x − 6 2 x + 6 y = 19 table with 2 rows and 1 column , row1 column 1 , y equals , 1 half , x minus 6 , row2 column 1 , 2 x plus 6 y equals 19 , end table

Task 2

Solve. Show all your work and explain your steps.

The triangle on the left has a perimeter of 14. The triangle on the right has a perimeter of 21. What are x and y?

BIG idea Modeling

You can represent many real-world mathematical problems algebraically. When you need to find two unknowns, you may be able to write and solve a system of equations or inequalities.

Task 3

Solve the problem. Show all of your work and explain your steps.

A town is organizing a Fourth of July parade. There will be two sizes of floats in the parade, as shown below. A space of 10 ft will be left after each float.

1. The parade must be at least 150 ft long, but less than 200 ft long. What combinations of large and small floats are possible?
2. Large floats cost $600 to operate. Small floats cost $300 to operate. The town has a budget of $2500 to operate the floats. How does this change your answer to part (a)? What combinations of large and small floats are possible?

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