Question

khalil is working two summer jobs, making $18 per hour tutoring and $9 per hour landscaping. last week khalil worked a total of 14 hours and earned a total of $180. graphically solve a system of equations in order to determine the number of hours khalil worked tutoring last week, x, and the number of hours khalil worked landscaping last week, y.

Explanation:

Step1: Define variables and set up equations

Let \( x \) be the hours tutoring, \( y \) be the hours landscaping.
Total hours: \( x + y = 14 \) (Equation 1)
Total earnings: \( 18x + 9y = 180 \) (Equation 2)

Step2: Rewrite equations in slope - intercept form (\( y = mx + b \))

For Equation 1:
\( y = -x + 14 \)

• When \( x = 0 \), \( y = 14 \); when \( x = 14 \), \( y = 0 \).

For Equation 2:
Divide by 9: \( 2x + y = 20 \) → \( y = -2x + 20 \)

• When \( x = 0 \), \( y = 20 \); when \( x = 10 \), \( y = 0 \).

Step3: Graph the lines and find intersection

Plot \( y = -x + 14 \) (points: (0,14), (14,0))
Plot \( y = -2x + 20 \) (points: (0,20), (10,0))
The intersection point of the two lines is where \( x + y = 14 \) and \( 18x + 9y = 180 \) are both satisfied.

To find the intersection algebraically (to confirm):
Substitute \( y = 14 - x \) into \( 18x + 9y = 180 \):
\( 18x + 9(14 - x)=180 \)
\( 18x + 126 - 9x = 180 \)
\( 9x = 54 \)
\( x = 6 \)
Then \( y = 14 - 6 = 8 \)

Answer:

Khalil worked 6 hours tutoring (\( x = 6 \)) and 8 hours landscaping (\( y = 8 \)). The intersection point of the two lines \( y=-x + 14\) and \(y = - 2x+20\) is \((6,8)\), so the solution to the system is \(x = 6\), \(y = 8\).