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:

x{ a − b + ab/(a − b) } = y{ a + b − ab/(a − b) }  …… (1)

x + y = 2a²  …… (2)

Step 1: Simplify Equation (1)

Take LCM (a − b):

x{ (a − b)² + ab } / (a − b) = y{ (a + b)(a − b) − ab } / (a − b)

x( a² − ab + b² ) = y( a² − ab − b² )

⇒ (a² − ab + b²)x − (a² − ab − b²)y = 0  …… (1)

Step 2: Write Equations in Standard Form

(a² − ab + b²)x − (a² − ab − b²)y = 0  …… (1)

x + y − 2a² = 0  …… (2)

Step 3: Compare with ax + by + c = 0

From equation (1): a₁ = (a² − ab + b²), b₁ = −(a² − ab − b²), c₁ = 0

From equation (2): a₂ = 1, b₂ = 1, c₂ = −2a²

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

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

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

Step 5: Find the Values of x and y

x = (a² − ab − b²) / (a² + b²)

y = (a² − ab + b²) / (a² + b²)

Final Answer

∴ The solution of the given system of equations is:

x = (a² − ab − b²) / (a² + b²)
y = (a² − ab + b²) / (a² + b²)

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of linear equations is ( (a² − ab − b²) / (a² + b²), (a² − ab + b²) / (a² + b²) ).