Application problem with a linear function: Finding a coordinate given two points

Question

Chau is driving to Denver. Suppose that the distance to his destination (in miles) is a linear function of his total driving time (in minutes). Chau has 54 miles to his destination after 45 minutes of driving, and he has 38.7 miles to his destination after 63 minutes of driving. How many miles will he have to his destination after 73 minutes of driving?

Miles:

Answer 1

The standard format of a linear function is y = mx + b, where m is the coefficient of x and b is a constant. In this problem, x represents the total driving time in minutes, while y represents the the distance to Chau's destination. If we plug in the numbers provided in this problem, we get the following equations:

y = mx + b

54 = 45m + b

38.7 = 63m + b

What we now have is a system of equations, which will enable us to solve for the coefficient m and the constant b. To do so, we will subtract the bottom equation from the top equation (which will cancel out the b's):

54 = 45m + b

38.7 = 63m + b

____________

15.3 = -18m

We then divide both sides by -18 in order to solve for m:

15.3 / -18 = -18m / -18

m = -0.85

Next, we take our answer (-0.85) and plug it into either one of the equations in order to solve for b:

54 = 45m + b

54 = 45(-0.85) + b

54 = -38.25 + b (Add 38.25 to both sides in order to isolate b)

b = 92.25

To obtain our linear equation, we simply plug in the answers for m and b:

y = mx + b ----> y = -0.85x + 92.25

Now we can find out how many miles are left to Chau's destination by plugging into the equation the number of minutes he drove (which is 73):

y = -0.85x + 92.25

y = =0.85(73) + 92.25

y =30.2

Chau therefore has 30.2 more miles to go after 73 minutes of driving.