How do you solve the system #4x-y=1, x+2y=7# using matrix equation?

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Answer 1 · 2017-02-14T18:40:39

# ((x),(y)) = ((1),(3))#

In matrix form it's this:

#((4,-1),(1,2))((x),(y)) = ((1),(7))#

You can the go through the formality of getting the inverse of the matrix (assuming of course it has one). If we start with the determinant:

#det ((4,-1),(1,2)) = 4(2) - 1(-1) = 9# so the matrix is invertible!

And we know that the inverse is:

#((4,-1),(1,2))^(-1) = 1/9 ((2,1),(-1,4))#

And we can say that:

#color(red)( 1/9 ((2,1),(-1,4))) color(green)( ((4,-1),(1,2))) ((x),(y)) = color(red)( 1/9 ((2,1),(-1,4))) ((1),(7))#

And because the red and the green terms combine as the identity matrix:

#implies ((x),(y)) = 1/9 ((2,1),(-1,4)) ((1),(7))#

#implies ((x),(y)) = 1/9 ((9),(27)) = ((1),(3))#

Generally speaking solving systems is easier using Row Reduction but in the case of a 2x2 it is also pretty easy to work through this method.