To solve the system of equations given by:
- \( 5x + 2y = 14 \)
- \( x - 6y = 22 \)
We can use the substitution or elimination method. Here, we'll use the substitution method.
First, let's solve the second equation for \( x \):
\[ x = 6y + 22 \]
Now we can substitute \( x \) in the first equation:
\[ 5(6y + 22) + 2y = 14 \]
Expanding this:
\[ 30y + 110 + 2y = 14 \]
Combining like terms:
\[ 32y + 110 = 14 \]
Next, we isolate \( y \) by subtracting 110 from both sides:
\[ 32y = 14 - 110 \] \[ 32y = -96 \]
Now, divide by 32:
\[ y = -\frac{96}{32} = -3 \]
Now that we have \( y \), we can find \( x \) using our expression for \( x \):
\[ x = 6(-3) + 22 \] \[ x = -18 + 22 \] \[ x = 4 \]
So, the solution to the system of equations is:
\[ (x, y) = (4, -3) \]
Verification
Let's verify by substituting \( x = 4 \) and \( y = -3 \) back into the original equations.
- For \( 5x + 2y = 14 \):
\[ 5(4) + 2(-3) = 20 - 6 = 14 \quad \text{(True)} \]
- For \( x - 6y = 22 \):
\[ 4 - 6(-3) = 4 + 18 = 22 \quad \text{(True)} \]
Both equations are satisfied. Thus, the final answer is:
\[ \boxed{(4, -3)} \]