Question

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

5x + 7y = 31, 7x + 5y = 29

Answer 1

Given the system of simultaneous linear equations:

5x + 7y = 31

7x + 5y = 29

Step-wise calculation:

Rewrite the equations in the form:

5x + 7y – 31 = 0

⇒ a₁ = 5, b₁ = 7, c₁ = –31

7x + 5y – 29 = 0

⇒ a₂ = 7, b₂ = 5, c₂ = –29

Using the formula for cross-multiplication:

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

Calculate each part:

b₁c₂ – b₂c₁ = 7 × (–29) – 5 × (–31)

b₁c₂ – b₂c₁ = –203 + 155

b₁c₂ – b₂c₁ = –48

c₁a₂ – c₂a₁ = (–31) × 7 – (–29) × 5

c₁a₂ – c₂a₁ = –217 + 145

c₁a₂ – c₂a₁ = –72

a₁b₂ – a₂b₁ = 5 × 5 – 7 × 7

a₁b₂ – a₂b₁ = 25 – 49

a₁b₂ – a₂b₁ = –24

Thus, `x/(-48) = y/(-72) = 1/(-24)`.

Solve for (x) and (y):

`x = (-48)/(-24)`

x = 2

`y = (-72)/(-24)`

y = 3

The solution to the system is x = 2, y = 3.