# Solve the following system of equations using matrices. 2x + 4y = 16 -4x – 8y = 32

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Solve the system using matrices:

2x+4y=16

-4x-8y=32

There are a number of ways to solve using matrices:

(1) Cramer's method:

`x=|[16,4],[32,-8]|/|[2,4],[-4,-8]|`

But the denominator is `|[2,4],[-4,-8]|=2(-8)-(-4)(4)=0` . Since you cannot divide by zero there is no solution.

(2) Using inverse matrices:

Rewrite as `([2,4],[-4,-8])([x],[y])=([16],[32])`

To solve AX=B we multiply by the inverse matrix `X=A^(-1)B`

The inverse of a 2x2 matrix can be found by :

`A=([a,b],[c,d]) ==> A^(-1)=1/(ad-bc)([d,-b],[-c,a])` if it exists.

In this case ad-bc is 2(-8)-(-4)(4)=0, and since we cannot divide by zero there is no solution.

(3) We can row reduce the augmented matrix:

`([2,4,16],[-4,-8,32])` Multiply row 1 by 1/2 and row 2 by 1/4

`=([1,2,8],[-1,-2,8])` Replace row 2 with the sum of row 1 and row 2

`=([1,2,8],[0,0,16])`

This is in row reduced form, but the last row indicates that there are no solutions.

The graphs:

The lines are parallel.