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 + y/b = 2  …… (1)

ax − by = a² − b²  …… (2)

Step 1: Convert into Linear Equations

Multiply equation (1) by ab:

bx + ay = 2ab  …… (1)

Equation (2) is:

ax − by = a² − b²  …… (2)

Step 2: Write Equations in Standard Form

bx + ay − 2ab = 0  …… (1)

ax − by − (a² − b²) = 0  …… (2)

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

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

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

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

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

x / [ −a(a² + b²) ] = y / [ −b(a² + b²) ] = 1 / [ −(a² + b²) ]

Step 5: Find the Values of x and y

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

⇒ x = a

y / [ −b(a² + b²) ] = 1 / [ −(a² + b²) ]

⇒ y = b

Final Answer

∴ The solution of the given system of equations is:

x = a and y = b

Conclusion

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