Question

Using cross-multiplication method, solve the following system of simultaneous linear equations:

3x – 7y = 2, 4x – 3y = 9

Answer 1

Given:

3x – 7y = 2

4x – 3y = 9

Rewrite equations in the form (a₁x + b₁y + c₁ = 0) and (a₂x + b₂y + c₂ = 0):

3x – 7y – 2 = 0   ...(Equation 1)

4x – 3y – 9 = 0   ...(Equation 2)

Identify coefficients:

a₁ = 3, b₁ = –7, c₁ = –2

a₂ = 4, b₂ = –3, c₂ = –9

Using the cross-multiplication formula:

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

Calculate each term:

1. b₁c₂ – b₂c₁ = (–7)(–9) – (–3)(–2)

b₁c₂ – b₂c₁ = 63 – 6

b₁c₂ – b₂c₁ = 57

2. c₁a₂ – c₂a₁ = (–2)(4) – (–9)(3)

c₁a₂ – c₂a₁ = –8 + 27

c₁a₂ – c₂a₁ = 19

3. a₁b₂ – a₂b₁ = 3(–3) – 4(–7)

a₁b₂ – a₂b₁ = –9 + 28

a₁b₂ – a₂b₁ = 19

So:

`x/57 = y/19 = 1/19`

Equating the fractions:

`x = 57/19`

x = 3

`y = 19/19`

y = 1