10. Solve using Cramer's rule
Each unknown will be the quotient of the determinant obtained by substituting the answers in the right sides of the equations for the coefficients of the unknown divided by the determinant formed by taking the coefficients on the left sides of the equations.
and
To evaluate 3x3 determinants, we reduce the problem down to finding 2x2 determinants. Let us start with the determinant which is in all of the bottoms.
First make up a 3x3 checkerboard array of + and - signs.
Start in the upper left with a + and alternate signs going in both directions.
Next choose any row or any column. In our example, let us choose the third row. Since it doesn't make any difference which row or column we choose, the third row has more negative entries, and so will be more instructive as an example. Multiply each entry in the row or column by the 2x2 determinant you get by throwing out the row and column in which we find the entry.
If the entry comes from a positive position in the checkerboard array, add the product. If it comes from a negative position, subtract the product.
This gives us
Let us check our result by expanding by the middle column.
You can show that no matter which row or column you use in the expansion, you will get the same answer. It is important to find the determinant in the bottom first, because if it is zero, this method will not work. There will be some kind of parallelism happening with the planes in the graph, and there will not be a unique solution. Now that we have the bottoms in the computation for all three unknowns, let us expand the top determinants in each case by the top row.
and we get the same answer we got with the substitution method, the addition method, row operations, and the inverse method, and that answer will check.