Writing and Rearranging Equations in Two Variables

1.9.3 Writing and Rearranging Equations in Two Variables

1.9.3 • Writing and Rearranging Equations in Two Variables

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.

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

Compare your answer:

$84, 000 m + 36, 000 p = 1, 962, 800$

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

Compare your answer:

$p = \frac{1, 962, 800 - 84, 000 m}{36, 000}$

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

Compare your answer:

It represents the number of new workers the department could hire if it resurfaces $m$ miles of roads and spends its entire budget.

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

Compare your answer:

$m = \frac{1, 962, 800 - 36, 000 p}{84, 000}$

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

Compare your answer:

It represents the miles of roads the department could resurface if it spends the entire budget and hires $p$ new workers.

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?

$m = \frac{1, 962, 800 - 36, 000 p}{84, 000}$
$84, 000 m + 36, 000 p = 1, 962, 800$
$p = \frac{1, 962, 800 - 84, 000 m}{36, 000}$

$m = \frac{1, 962, 800 - 36, 000 p}{84, 000}$

Explain your reasoning.

Compare your answer:

Substitute 6 for $p$ in the expression that is equal to $m$ to find the miles of roads that could be resurfaced.

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

20.8

$m = \frac{1, 962, 800 - 36, 000 (6)}{84, 000}$

$m = \frac{1, 962, 800 - 216, 00084, 000}{84, 000}$

$m = 20.8$

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.

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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?

$p= \frac{m}{4.00}$
$a= \frac{M-4.00p}{1.75}$
$M=4.00p+0.75a$
$p= \frac{M-0.75a}{4.00}$

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

Step 1 - Write the formula.

$d = r t$

Step 2 - Substitute.

$520 = 65 t$

Step 3 - Divide, to isolate $t$ .

$\frac{520}{65} = \frac{65 t}{65}$

Step 4 - Simplify.

$t = 8$

Example 2

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

Step 1 - Write the formula.

$d = r t$

Step 2 - Substitute.

(step not needed)

Step 3 - Divide, to isolate $t$ .

$\frac{d}{r} = \frac{r t}{r}$

Step 4 - Simplify.

$\frac{d}{r} = t$

Example 3

Solve the formula $A = \frac{1}{2} b h$ for $h$ , when $A = 90$ and $b = 15$ .

Step 1 - Write the formula.

$A = \frac{1}{2} b h$

Step 2 - Substitute.

$90 = \frac{1}{2} (15) h$

Step 3: Multiply to clear the fraction.

$2 \times 90 = 2 \times \frac{1}{2} (15) h$

Step 4 - Simplify.

$180 = 15 h$

Step 5 - Solve for $h$ .

$12 = h$

Example 4

Solve the formula $A = \frac{1}{2} b h$ for $h$ .

Step 1 - Write the formula.

$A = \frac{1}{2} b h$

Step 2 - Substitute.

(step not needed)

Step 3 - Multiply to clear the fraction.

$2 \times A = 2 \times \frac{1}{2} b h$

Step 4 - Simplify.

$2 A = b h$

Step 5 - Solve for $h$ .

$\frac{2 A}{b} = h$

Example 5

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

Step 1 - Write the formula.

$3 x + 2 y = 18$

Step 2 - Substitute.

$3 (4) + 2 y = 18$

Step 3 - Simplify.

$12 + 2 y = 18$

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

$12 - 12 + 2 y = 18 - 12$

Step 5 - Simplify.

$2 y = 6$

Step 6 - Divide.

$\frac{2 y}{2} = \frac{6}{2}$

Step 7 - Simplify.

$y = 3$

Example 6

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

Step 1 - Write the formula.

$3 x + 2 y = 18$

Step 2 - Substitute.

(step not needed)

Step 3 - Simplify.

(step not needed)

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

$3 x - 3 x + 2 y = 18 - 3 x$

Step 5 - Simplify.

$2 y = 18 - 3 x$

Step 6 - Divide.

$\frac{2 y}{2} = \frac{18 - 3 x}{2}$

Step 7 - Simplify.

$y = - \frac{3}{2} x + 9$

Try it

Solve a Formula for a Specific Variable

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

Compare your answer:

Here is how to solve this equation for $x$ by isolating:

Step 1 - Use inverse operations to bring the $y$ term to the right side of the equation by adding 4y. $2 x - 4 y + 4 y = 20 + 4 y$

Step 2 - Simplify. $2 x = 20 + 4 y$

Step 3 - Divide both sides by 2. $\frac{2 x}{2} = \frac{20 + 4 y}{2}$

Step 4 - Simplify. $x = 10 + 2 y$