To solve the augmented matrix:
\[ \begin{bmatrix} -3 & 1 & | & 4 \ 5 & 3 & | & 12 \end{bmatrix} \]
We can represent the equations based on the matrix:
- \(-3x + y = 4\)
- \(5x + 3y = 12\)
Let's solve this system of equations.
Step 1: Solve for \(y\) in the first equation
From the first equation, we can express \(y\) in terms of \(x\):
\[ y = 4 + 3x \]
Step 2: Substitute \(y\) into the second equation
Now we substitute this expression for \(y\) into the second equation:
\[ 5x + 3(4 + 3x) = 12 \]
Step 3: Simplify and solve for \(x\)
Expanding and simplifying:
\[ 5x + 12 + 9x = 12 \]
\[ 14x + 12 = 12 \]
Subtracting 12 from both sides:
\[ 14x = 0 \]
Dividing by 14 gives:
\[ x = 0 \]
Step 4: Back substitute to find \(y\)
Now we substitute \(x = 0\) back into the equation for \(y\):
\[ y = 4 + 3(0) = 4 \]
Solution
The solution to the system is:
\[ (x, y) = (0, 4) \]
Therefore, the final answer is:
\((0, 4)\)