Sometimes using matrix algebra or inverse matrices to find the solution to a system of linear equations can be tedious. Sometimes it is more convenient to use Cramer's Rule and determinants to solve a system of equations. Finding determinants becomes much more difficult with higher dimensions, and so Cramer's Rule is better for smaller systems of linear equation.
One thing you can do with determinants is solve systems of linear equations with them and the method's called Cramer's rule, so let's start with the system 9x+3y=12, 10x-4y=50 two equations, two unknowns. Cramer's rule says the solution will be x equals this determinant 12,3 50,-4 over the determinant 9,3 10,-4 now let me I may explain where these determinants come from. This determinant in the denominator is the determinant of the coefficient matrix right? 9, 3, 10, -4 that's from the coefficients on the left side. In the numerator, you basically have the same determinant only you've replaced the x coefficients with these numbers the constants so that's how you get x. You get y very similarly again in the denominator you've got the determinant of the coefficient matrix and in the numerator you've taken the coefficient matrix you've replaced the y terms with 12 and 50 and so it's very similar how you calculate this just remember replace the appropriate column with the constants in this case 12 and 50.
Let's actually calculate these and see what the solution is, so we're going to do x first. Let's observe that and you could still use the simplification rules for determinants whenever whenever possible like in the denominator I can pull a 3 out of this top row and I can pull a 2 out of the bottom row and that gives me 3 times 2 times the determinant 3,1 right? I pull a 3 out of the top I pull a 2 out of the bottom so I have 5 -2 and then in the in the top I can also pull out a 3 from the top row and a 2 from the bottom row see that's nice because I can actually cancel these factors of 3 and 2 and what's left is 4 1 and 25 -2 so as I said, you can just go ahead and cancel the 3 and the 2 and then on let's see the bottom first actually we get -6-5 that's -11 in the bottom and in the top, we get -8-25 that's -33 that's 3 so let's see the same thing for y again it's always a little bit easier if you can factor things out first because things factored out sometimes cancel so I'll pull 3 out and a 2 out again and I get 3,1 on top 5,-2 on the bottom and I can pull a 3 and a 2 at the top as well might as well just pull those out because I mean I can pull more out of the bottom obviously but there's no point these are going to cancel and nothing else will so let me just leave this as 3,4 and then I have 5,25 and then again the 6 cancels in the bottom I'm still going to have -6-5 -11 but in the top I'll have 75-20 55 and so that's 55 over -11 -5 so x=3, y=-5. This is Cramer's rule you're using determinants to solve a system of linear equations.