Question

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

2x – y = 9, 5x + y = 26

Answer 1

Given system of simultaneous linear equations:

2x – y = 9

5x + y = 26

Step 1: Write the equations in the form (a₁x + b₁y + c₁ = 0) and (a₂x + b₂y + c₂ = 0):

2⁢x – y – 9 = 0

⇒ a₁ = 2, b₁ = –1, c₁ = –9 

5x + y – 26 = 0

⇒ a₂ = 5, b₂ = 1, c₂ = –26

Step 2: Using 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 = (–1)(–26) – (1)(–9)

b₁c₂ – b₂c₁ = 26 + 9

b₁c₂ – b₂c₁ = 35

c₁a₂ – c₂a₁ = (–9)(5) – (–26)(2)

c₁a₂ – c₂a₁ = –45 + 52 

c₁a₂ – c₂a₁ = 7

a₁b₂ – a₂b₁ = (2)(1) – (5)(–1)

a₁b₂ – a₂b₁ = 2 + 5

a₁b₂ – a₂b₁ = 7

So,

`x/35 = y/7`

`x/35 = 1/7`

Step 3: Solve for (x) and (y):

`x = 35/7`

x = 5

`y = 7/7`

y = 1

The solution to the system is x = 5, y = 1.