Prove that the lines 2x – 3y + 1 = 0, x + y = 3, 2x – 3y = 2 and x

1.	Prove that the lines 2x – 3y + 1 = 0, x + y = 3, 2x – 3y = 2 and x + y = 4 form a parallelogram.
Answer» Given: 2x – 3y + 1 = 0, x + y = 3, 2x – 3y = 2 x + y = 4 are given equation To prove: The lines 2x – 3y + 1 = 0, x + y = 3, 2x – 3y = 2 and x + y = 4 form a parallelogram. Explanation: 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 proved, the given lines form a parallelogram.

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

Answer»

Given: 2x – 3y + 1 = 0,

x + y = 3,

2x – 3y = 2 x + y = 4 are given equation

To prove:

The lines 2x – 3y + 1 = 0, x + y = 3, 2x – 3y = 2 and x + y = 4 form a parallelogram.

Explanation:

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 proved, the given lines form a parallelogram.