Question

Prove that the lines 2x − 3y + 1 = 0, x + y = 3, 2x − 3y = 2  and x + y = 4 form a parallelogram.

Answer 1

The given lines can be written as

\[y = \frac{2}{3}x + \frac{1}{3}\]           ... (1)

\[y = - x + 3\]                  ... (2)

\[y = \frac{2}{3}x - \frac{2}{3}\]         ... (3) 

\[y = - x + 4\]                  ... (4)

The slope of lines (1) and (3) is \[\frac{2}{3}\] and that of lines (2) and (4) is −1.
Thus, lines (1) and (3), and (2) and (4) are two pair of parallel lines.
If both pair of opposite sides are parallel then ,we can say that it is a parallelogram.
Hence, the given lines form a parallelogram.