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 Question from Britney, a student: I need major help with this problem! Tomorrow is my semester test and I have no idea how to do this. 2x+5y=8 y-x=10

For a single 2x2 system the augmented matrix-row reduction method is overkill, but it's a useful technique. Other techniques are Cramer's rule, which is very slow for larger matrices but fast for 2x2, and simple row reduction which is faster if (as here) you are only looking for a single solution.

Rewrite your problems as a matrix equation (note that one equation has to be rearranged. Stacks of square brackets represent a single tall square bracket.):

[2 5][x]=[8 ]
[-1 1][y]=[10]

Then -1
[x] = [2 5] [8 ] (*)
[y] = [-1 1] [10]

and we must find the inverse matrix of

[2 5]
[-1 1] (I'm going to demonstrate with a different one.)

To do this we _augment_ it by sticking an identity matrix on the side

[3 2 | 1 0]
[-2 1 | 0 1]

Now use row operations to turn the left side into an identity matrix; this same sequence turns the right side into the inverse.

[1 2/3 | 1/3 0 ] (* 1/3)
[-2 1 | 0 1 ]

[1 2/3 | 1/3 0]
[0 7/3 | 2/3 1] (+ 2 r1)

[1 2/3 | 1/3 0]
[0 1 | 2/7 3/7] (* 3/7)

[1 0 | 1/7 -2/7] (-2/3 r2)
[0 1 | 2/7 3/7]

and the right side is what (if you are familiar with Cramer's rule) you would have expected. [But this method is far faster for a 3x3 or 4x4 matrix if you use the usual method to find the determinant!] Plug this into (*):

[x] = [1/7 -2/7] [8] = [-12/7]
[y] = [2/7 3/7] [10] = [ 46/7]

Now you do the same with your matrix!

Good Hunting!
RD

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