Question

justin and daniel work at a dry cleaners ironing shirts. justin can iron 40 shirts per hour, and daniel can iron 20 shirts per hour. daniel worked 6 more hours than justin and they ironed 360 shirts between them. graphically solve a system of equations in order to determine the number of hours justin worked, ( x ), and the number hours daniel worked, ( y ).

Explanation:

Step1: Define Variables and Equations

Let \( x \) be the hours Justin worked, \( y \) be the hours Daniel worked.
From "Daniel worked 6 more hours than Justin", we get:
\( y = x + 6 \) (Equation 1, a linear equation in slope - intercept form, slope \( m = 1 \), y - intercept \( b = 6 \))

From the total shirts ironed: Justin irons 40 shirts per hour, Daniel irons 20 shirts per hour, and total is 360. So:
\( 40x+20y = 360 \)
Simplify this equation: Divide both sides by 20, we get \( 2x + y=18 \), then \( y=- 2x + 18 \) (Equation 2, slope \( m=-2 \), y - intercept \( b = 18 \))

Step2: Graph Equation 1 (\( y=x + 6 \))

• When \( x = 0 \), \( y=0 + 6=6 \), so the point is \( (0,6) \).
• When \( x = 3 \), \( y=3 + 6 = 9 \), so the point is \( (3,9) \).

Plot these two points \((0,6)\) and \((3,9)\) and draw the line.

Step3: Graph Equation 2 (\( y=-2x + 18 \))

• When \( x = 0 \), \( y=-2(0)+18 = 18 \), so the point is \( (0,18) \).
• When \( x = 5 \), \( y=-2(5)+18=-10 + 18 = 8 \), so the point is \( (5,8) \).

Plot these two points \((0,18)\) and \((5,8)\) and draw the line.

Step4: Find Intersection Point

The intersection of the two lines \( y=x + 6 \) and \( y=-2x + 18 \) is the solution.
Set \( x + 6=-2x+18 \) (by equating the two equations, since at intersection \( y \) values are equal).
\( x+2x=18 - 6 \)
\( 3x = 12 \)
\( x = 4 \)
Substitute \( x = 4 \) into \( y=x + 6 \), we get \( y=4 + 6=10 \).

(Graphically, we look for the point where the two lines cross. From the equations' intersection, we confirm the solution.)

Answer:

Justin worked \( \boldsymbol{4} \) hours, Daniel worked \( \boldsymbol{10} \) hours.