Answer the following question:

Find the equation of the line which passes through the point of intersection of lines x + y + 9 = 0 , 2x + 3y + 1 = 0 and which makes X-intercept 1.

#### Solution

Let u ≡ x + y + 9 = 0 and v ≡ 2x + 3y + 1 = 0

Equation of the line passing through the point of intersection of lines u = 0 and v = 0 is given by u + kv = 0.

∴ (x + y + 9) + k(2x + 3y + 1) = 0 …(i)

∴ x + y + 9 + 2kx + 3ky + k = 0

∴ (1 + 2k)x + (1 + 3k)y + 9 + k = 0

But, x–intercept of this line is 1.

∴ `(-(9 + "k"))/(1 + 2"k")` = 1

∴ – 9 – k = 1 + 2k

∴ k = `(-10)/3`

Substituting the value of k in (i), we get

`(x + y + 9) + ((-10)/3) (2x + 3y + 1)` = 0

∴ 3(x + y + 9) – 10(2x + 3y + 1) = 0

∴ 3x + 3y + 27 – 20x – 30y – 10 = 0

∴ – 17x – 27y + 17 = 0

∴ 17x + 27y – 17 = 0, which is the equation of the required line.