Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Jada has time on the weekends to earn some money. A local bookstore is looking for someone to help sort books and will pay $12.20 an hour. To get to and from the bookstore on a work day, however, Jada would have to spend $7.15 on bus fare.

1.

Write an equation that represents Jada’s take-home earnings in dollars, $E$ , if she works at the bookstore for $h$ hours in one day.

Compare your answer:

$E = 12.20 h - 7.15$

2.

One day, Jada takes home $90.45 after working $h$ hours and after paying the bus fare. Write an equation to represent this situation.

Compare your answer:

$90.45 = 12.20 h - 7.15$ (or equivalent)

3.

Is 4 a solution to the last equation you wrote?

If so, be prepared to explain how you know it is a solution.
If not, be prepared to explain why it is not a solution. Then, find the solution.

Compare your answer:

Substituting 4 into the equation gives $90.45 = 12.20 (4) - 7.15$ or $90.45 = 41.65$ , which is not a true statement. The solution is 8. Substituting 8 into the equation gives $90.45 = 90.45$ , which is true.

4.

Is 7 a solution to the last equation you wrote?

If so, be prepared to explain how you know it is a solution.
If not, be prepared to explain why it is not a solution. Then, find the solution.

Compare your answer:

Substituting 7 into the equation gives $90.45 = 78.25$ , which is also false. The solution is 8. Substituting 8 into the equation gives $90.45 = 90.45$ , which is true.

5.

In this situation, what does the solution to the equation tell us?

Compare your answer:

It tells us the number of hours that Jada worked that allowed her to take home $90.45 after paying for her bus fare.

Are you ready for more?

Extending Your Thinking

Jada has a second option to earn money. She could help some neighbors with errands and computer work for $11 an hour. After reconsidering her schedule, Jada realizes that she has about 9 hours available to work one day of the weekend.

Which option should she choose, sorting books at the bookstore or helping her neighbors? Be prepared to show your reasoning.

Compare your answer: Your answer may vary, but here is a sample.

You could argue either way, depending on the assumptions you make.

Jada should work at the bookstore because she would earn more there. Her pay would be $109.80, and after subtracting $7.15 for the bus fare, she would still earn $102.65. She would earn $99 from the other option.

Jada should help her neighbors. Working 9 hours at the bookstore would mean a few dollars more than working 9 hours helping her neighbors, but it would also mean losing some personal time because of the travel involved.

Self Check

Jada found a third job option and was offered a position working at a camp for $10.50 per hour. The total bus fare to and from the camp is $8.40. Write an equation that represents how much she will make in a day,

d

, after working

h

hours.

$d=8.40h +10.50$
$d=\frac{10.50h}{8.40}$
$d = 10.50h + 8.40$
$d = 10.50h - 8.40$

Additional Resources

Verify a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same so that we end up with a true statement. Any value of the variable that makes the equation true is called a solution of an equation. It is the answer to the puzzle!

How to determine whether a number is a solution to an equation:

Step 1 - Substitute the number in for the variable in the equation.

Step 2 -Simplify the expressions on both sides of the equation

Step 3 -Determine whether the resulting equation is true (the left side is equal to the right side).

If it is true, the number is a solution.

If it is not true, the number is not a solution.

For example, the equation $2 l + 2 w = 72$ represents the relationship between the length, $l$ , and the width, $w$ , of a rectangle that has a perimeter of 72 units. If we know that the length is 15 units, what is the value of the width?

Step 1 - Substitute the number in for the variable in the equation.

$2 (15) + 2 w = 72$

This is an equation in one variable because $w$ is the only quantity that we don't know. To solve this equation means to find a value of $w$ that makes the equation true.

Step 2 -Simplify the expressions on both sides of the equation In this case, 21 is the solution because substituting 21 for $w$ in the equation results in a true statement.

$2 (15) + 2 w = 72$

$2 (15) + 2 (21) = 72$

$30 + 42 = 72$

Step 3 -Determine whether the resulting equation is true (the left side is equal to the right side).

If it is true, the number is a solution.

If it is not true, the number is not a solution.

$72 = 72$

Try it

Try It: Verify a Solution of an Equation

The equation $2 l + 2 w = 48$ represents the perimeter of a rectangle.

1.

If the width of the rectangle is 10, what is the new equation?

Compare your answer:

$2 l + 2 (10) = 48$ or $2 l + 20 = 48$

2.

What is the length of the rectangle that has a width of 10?

14

3.

Verify the solution.

Compare your answer:

To verify the solution, substitute $w = 10$ and $l = 14$ into $2 l + 2 w = 48$ . $2 (14) + 2 (10) = 48$ $28 + 20 = 48$ $48 = 48$ is a true statement, so the solution works!

1.4.2 Writing Equations to Represent Constraints

Activity

Are you ready for more?

Extending Your Thinking

Additional Resources

Verify a Solution of an Equation

Try It: Verify a Solution of an Equation