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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, m m , and the number of new workers it could hire, p p , if it spends the entire budget.

2.

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

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 m m .

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?

  1. m = 1 , 962 , 800 36 , 000 p 84 , 000 m = 1 , 962 , 800 36 , 000 p 84 , 000
  2. 84 , 000 m + 36 , 000 p = 1 , 962 , 800 84 , 000 m + 36 , 000 p = 1 , 962 , 800
  3. p = 1 , 962 , 800 84 , 000 m 36 , 000 p = 1 , 962 , 800 84 , 000 m 36 , 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: Rewriting Equations for a Specific Variable

Watch the following video for more help on how to write an equation for a specific variable and why it is helpful to do so.

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 = r t d = r t for t t when d = 520 d = 520 and r = 65 r = 65 .

Step 1 - Write the formula.

d = r t d = r t

Step 2 - Substitute.

520 = 65 t 520 = 65 t

Step 3 - Divide, to isolate t t .

520 65 = 65 t 65 520 65 = 65 t 65

Step 4 - Simplify.

t = 8 t = 8

Example 2

Solve the formula d = r t d = r t for t t .

Step 1 - Write the formula.

d = r t d = r t

Step 2 - Substitute.

(step not needed)

Step 3 - Divide, to isolate t t .

d r = r t r d r = r t r

Step 4 - Simplify.

d r = t d r = t

Example 3

Solve the formula A = 1 2 b h A = 1 2 b h for h h , when A = 90 A = 90 and b = 15 b = 15 .

Step 1 - Write the formula.

A = 1 2 b h A = 1 2 b h

Step 2 - Substitute.

90 = 1 2 ( 15 ) h 90 = 1 2 ( 15 ) h

Step 3: Multiply to clear the fraction.

2 × 90 = 2 × 1 2 ( 15 ) h 2 × 90 = 2 × 1 2 ( 15 ) h

Step 4 - Simplify.

180 = 15 h 180 = 15 h

Step 5 - Solve for h h .

12 = h 12 = h

Example 4

Solve the formula A = 1 2 b h A = 1 2 b h for h h .

Step 1 - Write the formula.

A = 1 2 b h A = 1 2 b h

Step 2 - Substitute.

(step not needed)

Step 3 - Multiply to clear the fraction.

2 × A = 2 × 1 2 b h 2 × A = 2 × 1 2 b h

Step 4 - Simplify.

2 A = b h 2 A = b h

Step 5 - Solve for h h .

2 A b = h 2 A b = h

Example 5

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

Step 1 - Write the formula.

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

Step 2 - Substitute.

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

Step 3 - Simplify.

12 + 2 y = 18 12 + 2 y = 18

Step 4 - Isolate the y y -term using subtraction.

12 12 + 2 y = 18 12 12 12 + 2 y = 18 12

Step 5 - Simplify.

2 y = 6 2 y = 6

Step 6 - Divide.

2 y 2 = 6 2 2 y 2 = 6 2

Step 7 - Simplify.

y = 3 y = 3

Example 6

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

Step 1 - Write the formula.

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

Step 2 - Substitute.

(step not needed)

Step 3 - Simplify.

(step not needed)

Step 4 - Isolate the y y -term using subtraction.

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

Step 5 - Simplify.

2 y = 18 3 x 2 y = 18 3 x

Step 6 - Divide.

2 y 2 = 18 3 x 2 2 y 2 = 18 3 x 2

Step 7 - Simplify.

y = 3 2 x + 9 y = 3 2 x + 9

Try it

Solve a Formula for a Specific Variable

Solve 2 x 4 y = 20 2 x 4 y = 20 for x x .

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