Question

How do you solve 9a + 7b = -30, 8b + 5c = 11, -3a + 10c = 73?

Answer 1

I have shown how to find `a=-1` using Kramer's rule in linear matrix algebra. The other 2 unknowns may be found in a similar fashion and I leave it as an exercise. :)

Explanation:

This is a system of 3 linear equations in 3 unknowns, so the easiest will be to use linear matrix algebra to solve it. (either Gaussian elimination, or the inverse matrix method, or Kramer's Rule).

I will use Kramer's Rule as it is probably the quickest.

First write the system of linear equations in coefficient matrix form :

`[(9,7,0),(0,8,5),(-3,0,10)]`

Now use co-factor expansion along any row or column of your choice to find the determinant of this matrix.
Proceeding along row 1, I get
`Delta=9(80-0)-7(0+15)+0=615`

We now form the 3 matrices by replacing each column vector in the coefficient matrix by the column vector of solutions and find the corresponding determinants in each case.

`Delta_a=|(-30,7,0),(11,8,5),(73,0,10)|=(-30)(80)-7(110-365)+0=-615`

`therefore` by Kramer's Rule, `a=(Delta_a)/Delta=(-615)/615=-1`

Proceeding in this fashion, we eventually find b and c. I leave the details as an exercise.