To solve the system of equations, let's start with the first pair of equations given:
- \( x + 27 = 14 \)
- \( y = 3x - 14 \)
Step 1: Solve for \(x\)
From the first equation, we can isolate \(x\):
\[ x + 27 = 14 \implies x = 14 - 27 \implies x = -13 \]
Step 2: Solve for \(y\)
Now we substitute \(x = -13\) into the second equation to find \(y\):
\[ y = 3(-13) - 14 = -39 - 14 = -53 \]
So from the first pair, we have:
\[ x = -13 \quad \text{and} \quad y = -53 \]
Now let's move on to the second pair of equations:
- \( 15x + 5y = 20 \)
- \( y = 8 - 5x \)
Step 3: Solve the second pair of equations
We can substitute equation 4 into equation 3:
\[ 15x + 5(8 - 5x) = 20 \]
Step 4: Distributing and solving
Now we distribute \(5\):
\[ 15x + 40 - 25x = 20 \]
Combine like terms:
\[ -10x + 40 = 20 \]
Now, isolate \(x\):
\[ -10x = 20 - 40 \implies -10x = -20 \implies x = 2 \]
Step 5: Solve for \(y\)
Now substitute \(x = 2\) back into equation 4 to solve for \(y\):
\[ y = 8 - 5(2) = 8 - 10 = -2 \]
So from the second pair, we have:
\[ x = 2 \quad \text{and} \quad y = -2 \]
Final Results:
From the first pair of equations:
- \(x = -13\)
- \(y = -53\)
From the second pair of equations:
- \(x = 2\)
- \(y = -2\)