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Algebra 1

1.9.3 Writing and Rearranging Equations in Two Variables

Algebra 11.9.3 Writing and Rearranging Equations in Two Variables

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Activity

The Department of Streets of a city has a budget of $1,962,800 for resurfacing roads and hiring additional workers this year.

The cost of resurfacing a mile of 2-lane road is estimated at $84,000. The average starting salary of a worker in the department is $36,000 a year.

Construction workers laying asphalt.
1.

Write an equation that represents the relationship between the miles of 2-lane roads the department could resurface, mm, and the number of new workers it could hire, pp, if it spends the entire budget.

2.

Use the equation you wrote in the first question to solve for pp.

3.

Explain what the solution to the equation in question 2 represents in this situation.

4.

Use the equation you wrote in the first question to solve for mm.

5.

Explain what the solution to the equation in question 4 represents in this situation.

6.

The city is planning to hire 6 new workers and use its entire budget. Which equation should be used to find how many miles of 2-lane roads it could resurface?

  • m=1,962,80036,000p84,000m=1,962,80036,000p84,000

  • 84,000m+36,000p=1,962,80084,000m+36,000p=1,962,800

  • p=1,962,80084,000m36,000p=1,962,80084,000m36,000

7.

Explain your reasoning.

8.

To the nearest hundredth, how many miles of 2-lane roads could the city resurface if it hires 6 new workers?

Video: Writing an Inequality for the Constraint

Watch the following video for more help solving this problem.

Self Check

A farmer makes money selling pumpkins and apples in the fall. His revenue, M, is made from selling pumpkins, p, for $4.00 and apples, a, for $0.75. Which equation is solved to help him find out how many pumpkins he sold?
  1. p = m 4.00
  2. a = M 4.00 p 1.75
  3. M = 4.00 p + 0.75 a
  4. p = M 0.75 a 4.00

Additional Resources

Solve a Formula for a Specific Variable

Solving for a variable in a formula is actually finding equivalent equations.

To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of 1. All other variables and constants are on the other side of the equals sign. To see how to solve a formula for a specific variable, we will start with the distance, rate and time formula.

Example 1

Solve the formula d=rtd=rt for tt when d=520d=520 and r=65r=65.

Step 1 - Write the formula.

d=rtd=rt

Step 2 - Substitute.

520=65t520=65t

Step 3 - Divide, to isolate tt.

52065=65t6552065=65t65

Step 4 - Simplify.

t=8t=8

Example 2

Solve the formula d=rtd=rt for tt.

Step 1 - Write the formula.

d=rtd=rt

Step 2 - Substitute.

(step not needed)

Step 3 - Divide, to isolate tt.

dr=rtrdr=rtr

Step 4 - Simplify.

dr=tdr=t

Example 3

Solve the formula A=12bhA=12bh for hh, when A=90A=90 and b=15b=15.

Step 1 - Write the formula.

A=12bhA=12bh

Step 2 - Substitute.

90=12(15)h90=12(15)h

Step 3: Multiply to clear the fraction.

2×90=2×12(15)h2×90=2×12(15)h

Step 4 - Simplify.

180=15h180=15h

Step 5 - Solve for hh.

12=h12=h

Example 4

Solve the formula A=12bhA=12bh for hh.

Step 1 - Write the formula.

A=12bhA=12bh

Step 2 - Substitute.

(step not needed)

Step 3 - Multiply to clear the fraction.

2×A=2×12bh2×A=2×12bh

Step 4 - Simplify.

2A=bh2A=bh

Step 5 - Solve for hh.

2Ab=h2Ab=h

Example 5

Solve the formula 3x+2y=183x+2y=18 for yy when x=4x=4.

Step 1 - Write the formula.

3x+2y=183x+2y=18

Step 2 - Substitute.

3(4)+2y=183(4)+2y=18

Step 3 - Simplify.

12+2y=1812+2y=18

Step 4 - Isolate the yy-term using subtraction.

1212+2y=18121212+2y=1812

Step 5 - Simplify.

2y=62y=6

Step 6 - Divide.

2y2=622y2=62

Step 7 - Simplify.

y=3y=3

Example 6

Solve the formula 3x+2y=183x+2y=18 for yy.

Step 1 - Write the formula.

3x+2y=183x+2y=18

Step 2 - Substitute.

(step not needed)

Step 3 - Simplify.

(step not needed)

Step 4 - Isolate the yy-term using subtraction.

3x3x+2y=183x3x3x+2y=183x

Step 5 - Simplify.

2y=183x2y=183x

Step 6 - Divide.

2y2=183x22y2=183x2

Step 7 - Simplify.

y=32x+9y=32x+9

Try it

Try It: Solve a Formula for a Specific Variable

Solve 2x4y=202x4y=20 for xx.

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