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, commax, and the number of hours Khalil worked landscaping last week, yy.

Question

give detailed instructions on how to graph this

Answer 1

To graphically solve the system of equations based on Khalil's work situation, we need to set up the equations first, then plot them graphically. Here is a detailed step-by-step process:

Step 1: Formulate the Equations

1. Define variables:

   • Let \( x \) = the number of hours spent tutoring.
   • Let \( y \) = the number of hours spent landscaping.

2. Set up the equations: From the problem, we have the following two relationships:

   • Equation 1 (total hours worked): \[ x + y = 14 \]

   • Equation 2 (total earnings): \[ 18x + 9y = 180 \]

Step 2: Simplify Equation 2

We can simplify Equation 2 (the total earnings equation) by dividing all terms by 9:

\[ 2x + y = 20 \]

Step 3: Rewrite the Equations in Slope-Intercept Form

We can rewrite both equations in the form \( y = mx + b \) for easier graphing.

1. From Equation 1: \[ y = 14 - x \]

2. From Equation 2: \[ y = 20 - 2x \]

Step 4: Prepare to Graph

1. Determine the intercepts for both equations for easier plotting:

   • For \( y = 14 - x \):

     • If \( x = 0 \): \( y = 14 \) (y-intercept)
     • If \( y = 0 \): \( x = 14 \) (x-intercept)

   • For \( y = 20 - 2x \):

     • If \( x = 0 \): \( y = 20 \) (y-intercept)
     • If \( y = 0 \): \( 2x = 20 \Rightarrow x = 10 \) (x-intercept)

2. Choose a graphing tool: You can use graph paper, a graphing calculator, or online graphing software like Desmos.

Step 5: Graph the Equations

1. Draw the axes:

   • Label the x-axis (representing hours tutoring, \( x \)) and y-axis (representing hours landscaping, \( y \)).
   • Choose sensible ranges for \( x \) and \( y \) based on the context (for example, \( 0 \) to \( 14 \) for both axes).

2. Plot the first line \( y = 14 - x \):

   • Plot the points: (0, 14) and (14, 0).
   • Draw a straight line through these points.

3. Plot the second line \( y = 20 - 2x \):

   • Plot the points: (0, 20) and (10, 0).
   • Draw a straight line through these points.

Step 6: Find the Intersection Point

Look for the point where the two lines intersect. This point represents the solution to the system of equations, indicating the values of \( x \) and \( y \).

Step 7: Interpret the Intersection Point

Once you find the intersection point:

• Read the coordinates of this point as \( (x, y) \).
• This will tell you the number of hours Khalil worked tutoring and landscaping.

Example Result

For instance, if the intersection is at \( (8, 6) \):

• Khalil worked 8 hours tutoring and 6 hours landscaping.

Conclusion

By following these steps, you can graphically find the solution to the system of equations relating to Khalil's work hours and earnings. Make sure to check your graph for accuracy to confirm the point of intersection.