Example 10 from Algebra and Trigonometry, 11.6 Solving Systems with Gaussian Elimination

Applying 2 × 2 Matrices to Finance

Carolyn invests a total of $12,000 in two municipal bonds, one paying 10.5% interest and the other paying 12% interest. The annual interest earned on the two investments last year was $1,335. How much was invested at each rate?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We have a system of two equations in two variables. Let $x =$ the amount invested at 10.5% interest, and $y =$ the amount invested at 12% interest.

$\begin{array}{l} x + y = 12,000 \\ 0.105 x + 0.12 y = 1,335 \end{array}$
As a matrix, we have

$[\begin{array}{r} 1 & 1 \\ 0.105 & 0.12 \end{array} | \begin{array}{r} 12,000 \\ 1,335 \end{array}]$
Multiply row 1 by $−0.105$ and add the result to row 2.
$[\begin{array}{r} 1 & 1 \\ 0 & 0.015 \end{array} | \begin{array}{r} 12,000 \\ 75 \end{array}]$
Then,

$\begin{array}{l} 0.015 y = 75 \\ y = 5,000 \end{array}$
So $12,000 −5,000 = 7,000.$
Thus, $5,000 was invested at 12% interest and $7,000 at 10.5% interest.

How did it go?

Solution

Solution