Question

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

3x + 4y = 25, 4x + 5y = 32

Answer 1

Given system of simultaneous linear equations:

3x + 4y = 25

4x + 5y = 32

Step 1: Rewrite in standard form (a₁x + b₁y + c₁ = 0): 

3x + 4y – 25 = 0

4x + 5y – 32 = 0 

Here, a₁ = 3, b₁ = 4, c₁ = –25 

a₂ = 4, b₂ = 5, c₂ = –32

Step 2: Apply 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 denominator:

a₁b₂ – a₂b₁ = 3 × 5 – 4 × 4

a₁b₂ – a₂b₁ = 15 – 16

a₁b₂ – a₂b₁ = –1

Calculate numerator for (x):

b₁c₂ – b₂c₁ = 4 × (–32) – 5 × (–25)

b₁c₂ – b₂c₁ = –128 + 125 

b₁c₂ – b₂c₁ = –3

Calculate numerator for (y):

c₁a₂ – c₂a₁ = (–25) × 4 – (–32) × 3

c₁a₂ – c₂a₁ = –100 + 96 

c₁a₂ – c₂a₁ = –4

Step 3: Compute values:

`x = (-3)/(-1)`

x = 3

`y = (-4)/(-1)`

y = 4