Question

Solve the following pair of equations by cross multiplication method.

2x − 5y = 14, x + 2y + 2 = 0

Answer 1

Given equations:

2x − 5y = 14

2x − 5y − 14 = 0   ...(1)

x + 2y + 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 = 2, b₁ = −5, c₁ = −14

a₂ = 1, b₂ = 2, 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

= (−5)(2) − (2)(14)

= −10 + 28

∴ b₁c₂ − b₂c₁ = 18

⇒ c₁a₂ − c₂a₁

= (−14)(1) − (2)(2)

= −14 − 4

∴ c₁a₂ − c₂a₁ = −18

⇒ a₁b₂ − a₂b₁

= (2)(2) − (1)(−5)

= 4 + 5

∴ a₁b₂ − a₂b₁ = 9

So, the value becomes,

`x/18 = y/-18 = 1/9`

Hence, finding x and y,

`x/18 = 1/9`

`x = (18)(1/9)`

`x = 18/9`

∴ x = 2

`y/-18 = 1/9`

`y = (-18)(1/9)`

`y = (-18)/9`

∴ y = −2

Thus, solving equations 2x − 5y = 14, x + 2y + 2 = 0 by cross multiplication method we get, x = 2 and y = −2.