Solving a tax rate or interest rate problem using a system of linear equations

Question

Lena bought a desktop computer and a laptop computer. Before finance charges, the laptop cost $250 more than the desktop. She paid for the computers using two different financing plans. For the desktop the interest rate was 8.5% per year, and for the laptop it was 5% per year. The total finance charges for one year were $296. 

How much did each computer cost before finance charges?

Desktop:

Laptop:

Answer 1

Hi Kristen!

To solve this problem, first, we need to be write variables for what we don't know. We can choose any letters, but it helps to write out what they are equal to.

L = price of laptop before financing charges; D = price of desktop before financing charges

Next, we need to be able to translate expressions.

" the laptop cost $250 more than the desktop" This means that the laptop, L, is 250 + the desktop, D, since "more than" means addition.

So L = 250 + D

Then, we need to understand how interest rates are applied to prices. Percentages are usually turned into decimals and then multiplied to the price,

Desktop was 8.5%---> 0.085 ----> desktop interest = 0.085D

Laptop was 5% ---> 0.05 ----> laptop interest = 0.05L

The interest from both the desktop and laptop added together was $296. (We know it will be adding together because the question said total).

So if desktop interest + laptop interest = $296, then 0.085D + 0.05L = $296

Now we have two equations

1. L = 250 + D
2. 0.085D + 0.05L = $296

that we can solve through either substitution or elimination.

Substitution is always easiest when 1) a variable has a coefficient of 1 and is easy to isolate or 2) if there is already a variable isolated on one side of the equation.

In this case, we have an equation with a variable already isolated - the L is isolated in the first equation. This means that we can now take 250+ D and plug it IN for "L" in the other equation.

L = 250 + D

0.085D + 0.05L = $296

0.085D + 0.05(250+D) = 296

0.085D + 12.5 + 0.05D = 296

0.135D + 12.5 = 296

-12.5 -12.5

----------------------------------

0.135D = 283.5

--------------------------

0.135

D = 2100

Now that we know that the price of the desktop is $2100, we can find the price of the laptop, L.

L = 250 + D ------> L = 250 + 2100 ----> L = 2350

We can double check our solutions into the second equation to see if they make sense.

0.085D + 0.05L = $296

0.085(2100) + 0.05(2350) = 296

178.5 + 117.5 = 296

296 = 296

Our answers are correct.

L = 2350 and D = 2100

Answer 2

Use the equation for simple interest for this problem... I = p*r*t

eq 1) I = p*.05 *1 = .05p

eq 2) I = (p + 250) * .085 = .085p + 21.25

Now add the two simplified equation and equal it to the total interest.

.05p + .085p + 21.25 = 296 then solve for p for which I leave for you to do.

Good luck to you.