Solve using augmented matrices: y = 4 and -6x + 6y = 18.

Steps to Solve

1. Convert equations into the standard form We have the equations: $$ y = 4 $$ and $$ -6x + 6y = 18 $$

Rearranging the first equation, we can express it in terms of both variables: $$ 0x + 1y = 4 $$

Now both equations can be written as:

1. $0x + 1y = 4$

2. $-6x + 6y = 18$

3. Create the augmented matrix We can represent the system of equations using an augmented matrix. The augmented matrix formed is: $$ \begin{pmatrix} 0 & 1 & | & 4 \ -6 & 6 & | & 18 \end{pmatrix} $$

4. Perform row operations To solve the system, we'll perform row operations. First, we can simplify the second row:

• We can divide the second row by 6: $$ R_2 \gets \frac{1}{6} R_2 $$

Now the augmented matrix looks like: $$ \begin{pmatrix} 0 & 1 & | & 4 \ -1 & 1 & | & 3 \end{pmatrix} $$

4. Eliminate one variable Next, we will eliminate the $y$ term in the second row. To do this, we can perform: $$ R_1 \gets R_1 + R_2 $$

The augmented matrix becomes: $$ \begin{pmatrix} -1 & 2 & | & 7 \ -1 & 1 & | & 3 \end{pmatrix} $$

5. Back substitution Now we can use back substitution. From the first row, we can express $x$ in terms of $y$: $$ -1x + 2y = 7 \implies x = 2y - 7 $$

From the second row: $$ -1x + 1y = 3 \implies x = y - 3 $$

6. Set the two expressions equal Now we equate the two expressions for $x$: $$ 2y - 7 = y - 3 $$

7. Solve for y Rearranging the equation gives: $$ 2y - y = 7 - 3 \implies y = 4 $$

8. Substitute y back to find x Now substitute $y = 4$ into one of the equations to find $x$: $$ x = 4 - 3 = 1 $$