Question

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

x – 3у – 5 = 0, 3x + 2y – 8 = 0

Answer 1

Given equations:

x – 3у – 5 = 0

3x + 2y – 8 = 0

Step 1: Identify coefficients

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

a₁ = 1, b₁ = –3, c₁ = –5 

a₂ = 3, b₂ = 2, c₂ = –8

Step 2: Apply 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:

b₁c₂ – b₂c₁ = (–3)(–8) – (2)(–5)

b₁c₂ – b₂c₁ = 24 + 10

b₁c₂ – b₂c₁ = 34

c₁a₂ – c₂a₁ = (–5)(3) – (–8)(1)

c₁a₂ – c₂a₁ = –15 + 8

c₁a₂ – c₂a₁ = –7

a₁b₂ – a₂b₁ = (1)(2) – (3)(–3)

a₁b₂ – a₂b₁ = 2 + 9

a₁b₂ – a₂b₁ = 11

Step 3: Write the ratios

`x/34 = y/(-7) = 1/11`

Step 4: Solve for (x) and (y)

`x = 34/11`

`y = (-7)/11`