Solve the System of Linear Equations Using Cross-Multiplication Method

Video Explanation

Watch the video below to understand the complete solution step by step:

Solution

Question: Solve the following system of equations using cross-multiplication method:

2(ax − by) + a + 4b = 0  …… (1)

2(bx + ay) + b − 4a = 0  …… (2)

Step 1: Simplify the Given Equations

2ax − 2by + a + 4b = 0

⇒ 2ax − 2by + (a + 4b) = 0  …… (1)

2bx + 2ay + b − 4a = 0

⇒ 2bx + 2ay + (b − 4a) = 0  …… (2)

Step 2: Compare with ax + by + c = 0

From equation (1): a₁ = 2a, b₁ = −2b, c₁ = (a + 4b)

From equation (2): a₂ = 2b, b₂ = 2a, c₂ = (b − 4a)

Step 3: Apply Cross-Multiplication Formula

x / (b₁c₂ − b₂c₁) = y / (a₂c₁ − a₁c₂) = 1 / (a₁b₂ − a₂b₁)

Substitute values:

x / [ (−2b)(b − 4a) − (2a)(a + 4b) ] = y / [ (2b)(a + 4b) − (2a)(b − 4a) ] = 1 / [ (2a)(2a) − (2b)(−2b) ]

x / [ −2b² + 8ab − 2a² − 8ab ] = y / [ 2ab + 8b² − 2ab + 8a² ] = 1 / [ 4a² + 4b² ]

x / [ −2(a² + b²) ] = y / [ 8(a² + b²) ] = 1 / [ 4(a² + b²) ]

Step 4: Find the Values of x and y

x / [ −2(a² + b²) ] = 1 / [ 4(a² + b²) ]

⇒ x = −1/2

y / [ 8(a² + b²) ] = 1 / [ 4(a² + b²) ]

⇒ y = 2

Final Answer

∴ The solution of the given system of equations is:

x = −1/2 and y = 2

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of linear equations is (−1/2, 2).