Question

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

x – 2y + 1 = 0, x + y + 4 = 0

Answer 1

Given the system of equations:

x – 2y + 1 = 0

x + y + 4 = 0

Step 1: Write coefficients

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

a₁ = 1, b₁ = –2, c₁ = 1 

a₂ = 1, b₂ = 1, c₂ = 4

Step 2: Calculate using the cross-multiplication formula

`x = (b_1c_2 - b_2c_1)/(a_1b_2 - a_2b_1)`

`y = (c_1a_2 - c_2a_1)/(a_1b_2 - a_2b_1)`

Calculate the denominator:

a₁b₂ – a₂b₁ = (1)(1) – (1)(–2)

a₁b₂ – a₂b₁ = 1 + 2

a₁b₂ – a₂b₁ = 3

Calculate numerator for (x):

b₁c₂ – b₂c₁ = (–2)(4) – (1)(1)

b₁c₂ – b₂c₁ = –8 – 1 

b₁c₂ – b₂c₁ = –9

Calculate numerator for (y):

c₁a₂ – c₂a₁ = (1)(1) – (4)(1)

c₁a₂ – c₂a₁ = 1 – 4

c₁a₂ – c₂a₁ = –3

Step 3: Substitute

`x = (-9)/3`

x = –3

`y = (-3)/3`

y = –1

The solution to the system by cross-multiplication method is x = –3, y = –1.