To solve the system of equations using substitution, we have the following equations:
- \( 5x + 2y = 14 \) (Equation 1)
- \( x - 6y = 22 \) (Equation 2)
Step 1: Solve one of the equations for one variable
Let's solve Equation 2 for \( x \):
\[ x = 6y + 22 \]
Step 2: Substitute this expression into the other equation
Now we substitute \( x \) in Equation 1:
\[ 5(6y + 22) + 2y = 14 \]
Step 3: Distribute and simplify
\[ 30y + 110 + 2y = 14 \]
Combine like terms:
\[ 32y + 110 = 14 \]
Step 4: Solve for \( y \)
Subtract 110 from both sides:
\[ 32y = 14 - 110 \]
\[ 32y = -96 \]
Now divide by 32:
\[ y = -3 \]
Step 5: Substitute \( y \) back to find \( x \)
Now, we substitute \( y = -3 \) back into the equation we derived for \( x \):
\[ x = 6(-3) + 22 \]
\[ x = -18 + 22 \]
\[ x = 4 \]
Final Solution
The solution to the system of equations is:
\((x, y) = (4, -3)\)
So the answer is:
\((4, -3)\)