Question

Solve the following pair of equations by cross multiplication method.

5x − 2y + 9 = 0, 4x + 3y = 2

Answer 1

Given equations:

5x − 2y + 9 = 0     ...(1)

4x + 3y = 2

4x + 3y − 2 = 0     ...(2)

Let’s write equations in standard form:

a₁x + b₁y₁ + c₁ = 0

a₂x + b₂y + c₂ = 0

Here, they are in the form of,

a₁ = 4, b₁ = −2, c₁ = 9

a₂ = 4, b₂ = 3, c₂ = −2

Using the identity:

`x/(b_1c_2 - b_2c_1) = y/(c_1a_2 - c_2a_1) = 1/(a_1b_2 - a_2b_1)`

Now, substituting the values,

⇒ b₁c₂ − b₂c₁

= (−2)(−2) − (3)(9)

= 4 − 27

∴ b₁c₂ − b₂c₁ = −23

⇒ c₁a₂ − c₂a₁

= (9)(4) − (−2)(5)

= 36 + 10

∴ c₁a₂ − c₂a₁ = 46

⇒ a₁b₂ − a₂b₁

= (5)(3) − (4)(−2)

= 15 + 8

∴ a₁b₂ − a₂b₁ = 23

So, the value becomes,

`x/-23 = y/46 = 1/23`

Hence, finding x and y,

`x/-23 = 1/23`

`x = (-23)(1/23)`

`x = (-23)/23`

∴ x = −1

`y/46 = 1/23`

`y = (46)(1/23)`

`y = 46/23`

∴ y = 2

Thus, solving equations 5x − 2y + 9 = 0, 4x + 3y = 2 by cross multiplication method we get, x = −1 and y = 2.