Systems of linear equation

Question

A bank loaned out $15,000, part of it at the rate of 13% annual interest, and the rest at 11% annual interest. The total interest earned for both loans was $1,720.00. How much was loaned at each rate?

Answer 1

Hey Ana C.

Setting up this question can be a little tricky. After it is set up, substitution is the simplest way to solve it.

First equation makes the most sense. There are two separate loans, x and y. We know how much was loaned out so

x + y =15,000

The second equation can be constructed from the later information. We can say that 11% annual interest is assigned to loan x and 13% annual interest is assigned to loan y( this is just the way I set it up). Converting the percents to decimals, we have our two equations:

x + y =15000

0.11x + 0.13y = 1720

Solving the first equation for y, we find

y= 15000 - x

and substitute for y in the second equation

0.11x + 0.13(15000 - x) = 1720

Solving for x we find

0.11x + 0.13(15000 - x) = 1720

0.11x+ 1950 - 0.13x = 1720 distribute 0.13 to 15000 and x

-0.02x = - 230 combine x terms and subtract 1950 from both sides

x = 11500 divide both sides by -0.02. Solved.

Now we substitute for x in our first equation

x + y = 15000

11500 + y = 15000 substitute for x

y =3500 subtract 11500 from both sides. Solved.

In conclusion, we have our first loan of $11500 and our second loan of $3500

-Pete