If the straight line passes through the point of intersection of the lines x + y = 3 and 2x – 3y = 1 and is parallel to x – y – 6 = 0, find a and b.

Given: lines are x + y = 3 and 2x − 3y = 1.

To find:

a and b.

Concept Used:

Point of intersection of two lines.

Explanation:

x + y − 3 = 0 … (1)

2x − 3y − 1 = 0 … (2)

Solving (1) and (2) using cross - multiplication method:

⇒ x = 2 , y = 1

Thus, the point of intersection of the given lines is (2, 1).

It is given that the line passes through (2, 1).

∴ ……(3)

It is also given that the line is parallel to the line x − y − 6 = 0.

Hence, Slope of is equal to the slope of x − y − 6 = 0 or, y = x – 6

∴ - b/a = 1

⇒ b = - a … (4)

From (3) and (4):

∴

⇒ a = 1

From (4):

b = −1

∴ a = 1,

b = −1

Hence, a = 1, b = - 1