Question

Do the equations \( x + 2y - 4 = 0 \) and \( 2x + 4y - 12 = 0 \) represent two parallel lines? Express this by geometrical method.

Hint:

When two lines have the same slope but different intercepts, they are parallel to each other.

Answer 1

The given equations are: \[ x + 2y - 4 = 0 \quad \text{(1)} \] and \[ 2x + 4y - 12 = 0 \quad \text{(2)}. \] First, simplify the second equation. Divide through by 2: \[ x + 2y - 6 = 0 \quad \text{(3)}. \] Now, compare equations (1) and (3): \[ x + 2y - 4 = 0 \quad \text{(1)} \] \[ x + 2y - 6 = 0 \quad \text{(3)}. \] Both equations have the same slope, since the coefficients of \(x\) and \(y\) are the same. However, their constant terms are different, which indicates that these are two parallel lines. Thus, the lines represented by the equations \( x + 2y - 4 = 0 \) and \( 2x + 4y - 12 = 0 \) are parallel to each other.
Conclusion:
The given equations represent two parallel lines.