Question

Solve the following pair of linear equations for \(x\) and \(y\) algebraically:
\(x + 2y = 9 \quad \text{and} \quad y - 2x = 2\)

Answer 1

Step 1: Formulate the system of equations:
The provided system is:
\[x + 2y = 9 \quad \text{(1)}\]
\[y - 2x = 2 \quad \text{(2)}\]

Step 2: Isolate a variable in one equation:
From equation (1), express \(x\) in terms of \(y\):
\[x = 9 - 2y \quad \text{(3)}\]

Step 3: Substitute into the other equation:
Substitute the expression for \(x\) from (3) into equation (2):
\[y - 2(9 - 2y) = 2\]
Simplify:
\[y - 18 + 4y = 2\]
\[5y - 18 = 2\]
Add 18 to both sides:
\[5y = 20\]
Divide by 5:
\[y = 4\]

Step 4: Back-substitute to find the other variable:
Substitute \(y = 4\) into equation (1) to find \(x\):
\[x + 2(4) = 9\]
\[x + 8 = 9\]
Subtract 8 from both sides:
\[x = 1\]

Step 5: Solution:
The system's solution is \(x = 1\) and \(y = 4\).