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: Write the system of equations:
We are given the following system of equations:
\[ x + 2y = 9 \quad \text{(1)} \] \[ y - 2x = 2 \quad \text{(2)} \]

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

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

Step 4: Substitute the value of \(y\) back into the first equation:
Now that we know \(y = 4\), substitute this value into equation (1) to solve for \(x\):
\[ x + 2(4) = 9 \] \[ x + 8 = 9 \] Subtract 8 from both sides:
\[ x = 1 \]

Step 5: Conclusion:
The solution to the system of equations is \(x = 1\) and \(y = 4\).