Question

Jace is working two summer jobs, making $18 per hour lifeguarding and

making $10 per hour washing cars. In a given week, he can work a maximum
of 13 total hours and must earn no less than $170. If x represents the number
of hours lifeguarding and y represents the number of hours washing cars,
write and solve a system of inequalities graphically and determine one
possible solution.

Answer 1

One possible solution is Jace spends 5 hours lifeguarding and 8 hours washing cars

Explanation:

The parameters of the two summer jobs Jace is working are;

The amount Jace makes for Lifeguarding = $18 per hour

The amount Jace makes for washing cars = $10 per hour

The number of hours he can work a week ≤ 13 hours

The amount he must earn in a ≥ $170

Whereby we have;

x = The number of hours Jace spends lifeguarding

y = The number of hours Jace spends washing cars

The system of inequalities are;

x × 18 + y × 10 ≥ 170

x + y ≤ 13

Making y the subject of the formula of both inequalities gives;

For the first inequality, we have;

x × 18 + y × 10 ≥ 170

10·y ≥ 170 - 18·x

y ≥ 170/10 - 18/10·x

y ≥ 17 - 1.8·x

For the second inequality, we have;

x + y ≤ 13

y ≤ 13 - x

Graphing both inequalities is drawn using Microsoft Excel

Equating both inequalities, given in relation to the y-values, the point where the two lines representing both inequalities meet is given as follows;

17 - 1.8·x = 13 - x

17 - 13 = 1.8·x - x

4 = 0.8·x

x = 4/0.8 = 5

y ≤ 13 - x gives;

y ≤ 13 - 5

y ≤ 8

Given that both relationships are either ≤ or ≥, we have the point where both inequalities meet is one possible solution

Therefore;

One possible solution is x = 5 hours, y = 8 hours

The number of hours Jace spends lifeguarding = x = 5 hours

The number of hours Jace spends washing cars = y = 8 hours.