How do you set up a linear equation based on a word problems?

Question

A car that weighs 2100 pounds gets 35 miles per gallon. A car that weighs 3000 pounds gets 27 miles per gallon. So if a car gets 25 miles per Gallon how much does the car weigh? How do I set up an equation based on the information given???

Answer 1

In this problem we're assuming that we can express weight (y) as a function of mileage (x), and we're given these values:

 

x[mpg]     y[lbs]

35            2100

27            3000

 

We're given two points and want to find the equation of the line connecting them (y=mx+b). Because we have the two points, we can figure out the slope (m) right away:

 

m = Δy/Δx = (y₂-y₁)/(x₂-x₁) = (3000-2100)/(27-35) = (900)/(-8) = -112.5.

 

Since we were given two points instead of the y-intercept (b), we need to construct the line using the point-slope formula. Notice that for this step it doesn't matter which of the two given xy-pairs we choose (I'll use the first pair, but check to make sure that the second pair gives you the same answer).

 

(y-y₀)=m(x-x₀)

 

(y-2100)=-112.5(x-35)

 

y-2100=-112.5x+3937.5

 

y=-112.5x+6037.5 

 

Now, to compute the weight of the 25 mpg car, we just substitute x=25 and solve for y:

 

y= -112.5(25)+6037.5 = -2812.5+6037.5 = 3225 lbs