5. Linda drove to Red Bluff. She drove 215 total miles. Part of the way she drove through the mountains where she could only average 50 mi/hr., and the rest of the way she drove on the Interstate through the valley where she was able to average 65 mi/hr. The trip took her 4 hours. How long was she driving through the mountains, and how long was she on the Interstate?

Again we start off by defining our unknowns. In this problemthey are asking for the time it took her to make these different parts of her journey so we will

let   x = her time through the mountains

y = her time on the interstate

We can then fill in the table as follows

Since   d = rt,   we can fill in the rest of the table as follows

When we have two unknowns, we need to have two equations. The first equation comes from the fact that the total time was 4 hours.

x + y = 4

The other one comes from the fact that the total distance was 215 miles.

50x + 65y = 215

If we solve the time equation for   x

x = 4 - y

we will get   x   expressed as a function of   y,   and we will be able to write the second equation using only one unknown

50(4 - y) + 65y = 215

Remove parentheses.

200 - 50y + 65y = 215

Combine like terms.

200 + 15y = 215

Subtract 200 from both sides.

15y = 15

Divide both sides by 15.

y = 1

So she spent 1 hour on the interstate. If her total time was 4 hours, she must have spent 3 hours in the mountains.

Check:

3 hours in the mountains at 50 mi/hr would cover 150 miles. 1 hour on the interstate at 65 mi/hr would have covered another 65 miles. Together they would have added up to 215 miles.

The numbers in the computations of the solution do have meaning. The 200 is how far she would have gotten if she had spent the entire 4 hours at 50 mi/hr. She didn't, and the fact that she was able to spend some time going faster allowed her to cover more miles. The 15 as the coefficient of the   y   is how much faster she was going, and the 15 on the other side of the equation is how much farther she was able to travel as a result of being able to go that much faster.