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:

(b/a)x + (a/b)y = a² + b²  …… (1)

x + y = 2ab  …… (2)

Step 1: Convert Equation (1) into Linear Form

Multiply equation (1) by ab:

b²x + a²y = ab(a² + b²)  …… (1)

Step 2: Write Equations in Standard Form

b²x + a²y − ab(a² + b²) = 0  …… (1)

x + y − 2ab = 0  …… (2)

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

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

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

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

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

x / [ −2a³b + ab(a² + b²) ] = y / [ −ab(a² + b²) + 2ab³ ] = 1 / ( b² − a² )

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

Step 5: Find the Values of x and y

x / [ ab(b² − a²) ] = 1 / ( b² − a² )

⇒ x = ab

y / [ ab(a² − b²) ] = 1 / ( b² − a² )

⇒ y = −ab

Final Answer

∴ The solution of the given system of equations is:

x = ab and y = −ab

Conclusion

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