Question

Solve the following pair of equations by cross multiplication method.

7x − y = 23, 8x + 3y = 18

Answer 1

Given equations:

7x − y = 23

7x − y − 23 = 0    ...(1)

8x + 3y = 18

8x + 3y − 18 = 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 = 7, b₁ = −1, c₁ = −23

a₂ = 8, b₂ = 3, c₂ = −18

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

= (−1)(−18) − (3)(−23)

= 18 + 69

∴ b₁c₂ − b₂c₁ = 87

⇒ c₁a₂ − c₂a₁

= (−23)(8) − (−18)(7)

= −184 + 126

∴ c₁a₂ − c₂a₁ = −58

⇒ a₁b₂ − a₂b₁

= (7)(3) − (8)(−1)

= 21 + 8

∴ a₁b₂ − a₂b₁ = 29

So, the value becomes,

`x/87 = y/-58 = 1/29`

Hence, finding x and y,

`x/87 = 1/29`

`x = (87)(1/29)`

`x = 87/29`

∴ x = 3

`y/-58 = 1/29`

`y = (-58)(1/29)`

`y = (-58)/29`

∴ y = −2

Thus, solving equations 7x − y = 23, 8x + 3y = 18 by cross multiplication method we get, x = 3 and y = −2.