Solve using augmented matrices. y = 9, -10x - 6y = -14

Steps to Solve

1. Set up the equations The given equations are:
2. $y = 9$
3. $-10x - 6y = -14$

We can rewrite the first equation in a standard form: $$0x + 1y = 9$$

So, our system of equations is:

• $0x + 1y = 9$
• $-10x - 6y = -14$

2. Write the augmented matrix The augmented matrix is formed by the coefficients of the variables and the constants: $$ \begin{bmatrix} 0 & 1 & | & 9 \ -10 & -6 & | & -14 \end{bmatrix} $$

3. Perform row operations We can start manipulating the rows to achieve row-echelon form.

First, to make the leading coefficient of the first row a 1, swap rows to position the row with the non-zero leading coefficient at the top: $$ \begin{bmatrix} -10 & -6 & | & -14 \ 0 & 1 & | & 9 \end{bmatrix} $$

Next, we can make the first row's leading coefficient a 1 by dividing the first row by -10: $$ \begin{bmatrix} 1 & \frac{3}{5} & | & \frac{7}{5} \ 0 & 1 & | & 9 \end{bmatrix} $$

4. Eliminate variables To eliminate the y-term in the first row, we can manipulate the second row: Multiply the second row by $\frac{3}{5}$ and subtract from the first row.

Performing this operation gives: $$ \begin{bmatrix} 1 & 0 & | & -\frac{32}{5} \ 0 & 1 & | & 9 \end{bmatrix} $$

5. Write the solutions From this augmented matrix, we can express the solutions:

• $x = -\frac{32}{5}$
• $y = 9$