# Write and row reduce the augmented matrix to find the general solution: x - 2y + 13 = 0, y - 4x =...

## Question:

Write and row reduce the augmented matrix to find the general solution: x - 2y + 13 = 0, y - 4x = 17.

## Augmented Matrix

One method that we can use in order to solve a system of equations is via a matrix. We can construct an augmented matrix by creating a matrix where each row represents an equation. The elements in each row are the coefficients on the variables, and then the final column is the number after the equals sign. Performing row operations will allow us to isolate a solution.

We can construct the augmented matrix by constructing the matrix where the coefficients on our variables are the elements in the matrix. The second equation needs to be rearranged so that x comes before y, and then the matrix can be found as follows.

{eq}\begin{bmatrix} 1 & -2 & -13\\ -4 & 1 & 17\end{bmatrix} {/eq}

Let's now perform row operations on this system to find the value of x and y that solve it.

{eq}\begin{align*} \begin{bmatrix} 1 & -2 & -13\\ -4 & 1 & 17\end{bmatrix} & R_2 = 4R_1 + R_2\\ \begin{bmatrix} 1 & -2 & -13\\ 0 & -7 & -35\end{bmatrix} & R_2 = -\frac{1}{7}R_2\\ \begin{bmatrix} 1 & -2 & -13\\ 0 & 1 & 5 \end{bmatrix} &R_1 = R_1 + 2R_2\\ \begin{bmatrix} 1 & 0 & -3\\ 0 & 1 & 5 \end{bmatrix} \end{align*} {/eq}

The solution therefore is {eq}x = -3, y = 5 {/eq}. 