Find the equation of the straight line joining the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4 and the point of intersection of the lines 2x + y – 4 = 0 and x – 2y + 3 = 0.

Here we have the straight line which joins the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4 and the point of intersection of the lines 2x + y – 4 = 0 and x – 2y + 3 = 0.

As the straight line which joins the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4,let us solve these 2 equations:

3x –y + 9 = 0–––(1)

X + 2y–4 = 0–––(2)

⇒ 3x –y + 9 = 0–––(1)

3x + 6y–12 = 0–––(2)(multiply (2) equation by 3 on both the sides)

Now,

(Subtract equation (2) from (1))

⇒ –7y = –21

⇒

Substitute y = 3 in the first equation

3x–y + 9 = 0

⇒ 3x–3 + 9 = 0

⇒ 3x + 6 = 0

⇒ 3x = –6

⇒

Thus, the point of intersection is(–2,3).

Now,

2x + y – 4 = 0–––(3)

x – 2y + 3 = 0–––(4)

(multiply (4) equation by 2 on both the sides) we get

2x – 4y + 6 = 0 --------(5)

Now,

⇒ 5y = 10

⇒

Substitute y = 2 in the (3) equation:

2x + y–4 = 0

⇒ 2x + 2–4 = 0

⇒ 2x–2 = 0

⇒ 2x = 2

⇒

The point of intersection is (1,2).

Hence equation of the line is

Here we have the points (–2,3) and (1,2)

Therefore,

⇒ 3(y–3) = –1(x + 2)

⇒ 3y–9 = –x–2

⇒ 3y–9 + x + 2 = 0

⇒ 3y + x–7 = 0

⇒ x + 3y–7 = 0