Question

Solve the following pair of equations by cross multiplication method.

9x + 5y = 7, 6x − y = 22

Answer 1

Given equations:

9x + 5y = 7

9x + 5y − 7 = 0   ...(1)

6x − y = 22

6x − y − 22 = 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 = 9, b₁ = 5, c₁ = −7

a₂ = 6, b₂ = −1, c₂ = −22

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

= (5)(−22) − (−1)(−7)

= −110 − 7

∴ b₁c₂ − b₂c₁ = −117

⇒ c₁a₂ − c₂a₁

= (−7)(6) − (−22)(9)

= −42 + 198

∴ c₁a₂ − c₂a₁ = 156

⇒ a₁b₂ − a₂b₁

= (9)(−1) − (6)(5)

= −9 − 30

∴ a₁b₂ − a₂b₁ = −39

So, the value becomes,

`x/-117 = y/156 = 1/-39`

Hence, finding x and y,

`x/-117 = 1/-39`

`x = (-117)(1/-39)`

`x = (-117)/-39`

∴ x = 3

`y/156 = 1/-39`

`y = (156)(1/-39)`

`y = 156/-39`

∴ y = −4

Thus, solving equations 9x + 5y = 7, 6x − y = 22 by cross multiplication method we get, x = 3 and y = −4.