Question

a rental car company charges a one time fee of $50 plus $1 per mile. another rental car company charges a one time fee of $10 plus $2 per mile. the equations at right represent the total cost, t, for renting a car for m miles at each company. for how many miles is the cost the same at both companies? t = m + 50 t = 2m + 10 rental car costs 1. graph the lines represented by the equations. 2. what is the point of intersection? 3. substitute the coordinates of the intersection point in each equation. do the coordinates represent a solution to both equations? t = m + 50 t = 2m + 10 □ ≟ □ + 50 □ ≟ 2(□) + 10 □ = □ □ ≟ □ + 10 □ = □ 4. for how many miles is the cost the same at both companies?

Explanation:

Step1: Set costs equal to each other

$m + 50 = 2m + 10$

Step2: Isolate m variable

$50 - 10 = 2m - m$

Step3: Solve for m

$40 = m$

Step4: Find total cost t

$t = 40 + 50 = 90$

Step5: Verify with second equation

$t = 2(40) + 10 = 90$

Answer:

1. (Graph instructions: For $t=m+50$, plot y-intercept $(0,50)$ and use slope 1 to plot another point like $(10,60)$, then draw the line. For $t=2m+10$, plot y-intercept $(0,10)$ and use slope 2 to plot another point like $(10,30)$, then draw the line.)
2. $(40, 90)$
3. For $t=m+50$:

$90 \stackrel{?}{=} 40 + 50$
$90 = 90$
For $t=2m+10$:
$90 \stackrel{?}{=} 2(40) + 10$
$90 \stackrel{?}{=} 80 + 10$
$90 = 90$
Yes, the coordinates represent a solution to both equations.

1. 40 miles