Question

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

4x + 3y = 5, 2x + 5y = –1

Answer 1

Given system of equations:

4x + 3y = 5

2x + 5y = –1

Step 1: Write equations in form (a₁x + b₁y + c₁ = 0) and (a₂x + b₂y + c₂ = 0) by bringing all terms to one side: 

4x + 3y – 5 = 0 

2x + 5y + 1 = 0

Here, a₁ = 4, b₁ = 3, c₁ = –5 and a₂ = 2, b₂ = 5, c₂ = 1.

Step 2: According to the cross-multiplication method, the variables (x) and (y) are given by the ratio:

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

Calculate each numerator and denominator:

b₁c₂ – b₂c₁ = 3 × 1 – 5 × (–5)

b₁c₂ – b₂c₁ = 3 + 25

b₁c₂ – b₂c₁ = 28

c₁a₂ – c₂a₁ = (–5) × 2 – 1 × 4

c₁a₂ – c₂a₁ = –10 – 4

c₁a₂ – c₂a₁ = –14

a₁b₂ – a₂b₁ = 4 × 5 – 2 × 3

a₁b₂ – a₂b₁ = 20 – 6

a₁b₂ – a₂b₁ = 14

Step 3: So, `x/28 = y/(-14) = 1/14`

Step 4: From this,

`x = 28/14`

x = 2 

`y = (-14)/(14)`

y = –1

The solution of the system is x = 2, y = –1.