# How do you set up and solve the following system using augmented matrices x-2y=7, 3x+4y=1?

Apr 30, 2017

#### Explanation:

The equation $x - 2 y = 7$ gives us the following line in the augmented matrix:

[ (1,-2,|,7) ]

The equation $3 x + 4 y = 1$ adds the second line to the augmented matrix:

[ (1,-2,|,7), (3,4,|,1) ]

Perform elementary row operation until an identity matrix is obtained on the left, then the column on the right will contain the solutions.

We want the coefficient in position $\left(1 , 1\right)$ to be 1 and it is, therefore, no operation is needed.

We want the other coefficient in column 1 to be zero, therefore, we perform the following row operation:

${R}_{2} - 3 {R}_{1} \to {R}_{2}$

[ (1,-2,|,7), (0,10,|,-20) ]

We want the coefficient is position $\left(2 , 2\right)$ to be one, therefore, we perform the following row operation:

${R}_{2} / 10 \to {R}_{2}$

[ (1,-2,|,7), (0,1,|,-2) ]

We want the other coefficient in column to be 0, therefore, we perform the following row operation:

${R}_{1} + 2 {R}_{2} \to {R}_{1}$

[ (1,0,|,3), (0,1,|,-2) ]

We have an identity matrix on the left, therefore the column on the right contains the solution: $x = 3 \mathmr{and} y = - 2$

Check:

$x - 2 y = 7$
$3 x + 4 y = 1$

$3 - 2 \left(- 2\right) = 7$
$3 \left(3\right) + 4 \left(- 2\right) = 1$

$7 = 7$
$1 = 1$

This checks.