Question

write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. a business owner wants to have the front window of her store painted and is considering two artists for the job. martina has quoted a flat rate of $85 for the job. reagan’s quote is $38 per hour, plus $47 to cover the cost of materials. depending on how long it takes reagan to paint the window, these two could end up charging the same amount. how long would reagan have to take? how much would it cost? if the job takes reagan □ hours, the cost would be $ □ with either artist.

Explanation:

Step1: Define variables and equations

Let \( h \) be the number of hours. Martina's cost equation: \( C_M = 85 + 47 \) (wait, no, Martina's quote is a flat rate? Wait, no, re-reading: Martina has quoted a flat rate of $85 for the job? Wait, no, "Martina has quoted a flat rate of $85 for the job. Reagan’s quote is $38 per hour, plus $47 to cover the cost of materials." So Martina's cost: \( C_M = 85 \) (flat rate). Reagan's cost: \( C_R = 38h + 47 \). We need to find \( h \) when \( C_M = C_R \).

Step2: Set equations equal

Set \( 85 = 38h + 47 \). Subtract 47 from both sides: \( 85 - 47 = 38h \). So \( 38 = 38h \).

Step3: Solve for h

Divide both sides by 38: \( h = \frac{38}{38} = 1 \).

Step4: Find the cost

Substitute \( h = 1 \) into either equation. Using Martina's: \( 85 \). Using Reagan's: \( 38(1) + 47 = 38 + 47 = 85 \).

Answer:

If the job takes Reagan \(\boxed{1}\) hours, the cost would be \(\boxed{85}\) with either artist.