Question

Find the equation of the line drawn through the point of intersection of the lines x + y = 9 and 2x – 3y + 7 = 0 and whose slope is −2/3.

Answer 1

Suppose the given two lines intersect at a point P(x₁, y₁). Then, (x₁, y₁) satisfies each of the given equations.

x + y = 9 …(i)

2x – 3y + 7 = 0 …(ii)

Now, we find the point of intersection of eq. (i) and (ii)

Multiply the eq. (i) by 2, we get

2x + 2y = 18

or 2x + 2y – 18 = 0 …(iii)

On subtracting eq. (iii) from (ii), we get

2x – 3y + 7 – 2x – 2y + 18 = 0

⇒ - 5y + 25 = 0

⇒ - 5y = -25

⇒ y = 5

Putting the value of y in eq. (i), we get

x + 5 = 9

⇒ x = 9 – 5

⇒ x = 4

Hence, the point of intersection P(x₁, y₁) is (4, 5)

Now, we have to find the equation of the line passing through the point (4, 5) and having slope = − 2/3

Equation of line: y – y₁ = m(x – x₁)

⇒ y - 5 = - 2/3(x - 4)

⇒ 3y - 15 = -2x+8

⇒ 2x + 3y – 15 – 8 = 0

⇒ 2x + 3y – 23 = 0

Hence, the equation of line having slope -2/3 is 2x + 3y – 23 = 0