A small shoe company took out a loan of $1,500,000 to expand their inventory. Part of the money was borrowed at 7%, part was borrowed at 8%, and part was borrowed at 10%. The amount borrowed at 10% was four times the amount borrowed at 7%, and the annual interest on all three loans was $130,500. Use matrices to find the amount borrowed at each rate.
An augmented matrix is one that contains the coefficients and constants of a system of equations. See
[link] .
A matrix augmented with the constant column can be represented as the original system of equations. See
[link] .
Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
We can use Gaussian elimination to solve a system of equations. See
[link] ,
[link] , and
[link] .
Row operations are performed on matrices to obtain row-echelon form. See
[link] .
To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions. See
[link] and
[link] .
A calculator can be used to solve systems of equations using matrices. See
[link] .
Many real-world problems can be solved using augmented matrices. See
[link] and
[link] .
Section exercises
Verbal
Can any system of linear equations be written as an augmented matrix? Explain why or why not. Explain how to write that augmented matrix.
Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar.
Is there only one correct method of using row operations on a matrix? Try to explain two different row operations possible to solve the augmented matrix
No, there are numerous correct methods of using row operations on a matrix. Two possible ways are the following: (1) Interchange rows 1 and 2. Then
(2)
Then divide row 1 by 9.
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