To solve the given augmented matrix, we will first rewrite it in a more standard form:
\[ \begin{bmatrix} 4 & 2 & | & 10 \ 1 & -1 & | & 13 \end{bmatrix} \]
This represents the system of equations:
- \( 4x + 2y = 10 \)
- \( x - y = 13 \)
We will solve this system step by step.
Step 1: Solve for one variable
From the second equation \( x - y = 13 \), we can express \( x \) in terms of \( y \): \[ x = y + 13 \]
Step 2: Substitute into the first equation
Now, we can substitute \( x \) into the first equation: \[ 4(y + 13) + 2y = 10 \] Expanding this: \[ 4y + 52 + 2y = 10 \] Combine like terms: \[ 6y + 52 = 10 \] Now, isolate \( y \): \[ 6y = 10 - 52 \] \[ 6y = -42 \] \[ y = -7 \]
Step 3: Substitute back to find \( x \)
Now that we have \( y \), we can substitute it back into the equation for \( x \): \[ x = y + 13 \] \[ x = -7 + 13 \] \[ x = 6 \]
Final Solution
Thus, the solution to the system is: \[ x = 6 \] \[ y = -7 \]
So, in conclusion: \[ x = 6, \quad y = -7 \]