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:

ax + by = a − b  …… (1)

bx − ay = a + b  …… (2)

Step 1: Write Equations in Standard Form

ax + by − (a − b) = 0  …… (1)

bx − ay − (a + b) = 0  …… (2)

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

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

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

Step 3: Apply Cross-Multiplication Formula

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

Substitute values:

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

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

x / ( −(a² + b²) ) = y / ( a² + b² ) = 1 / ( −(a² + b²) )

Step 4: Find the Values of x and y

x / ( −(a² + b²) ) = 1 / ( −(a² + b²) )

⇒ x = 1

y / ( a² + b² ) = 1 / ( −(a² + b²) )

⇒ y = −1

Final Answer

∴ The solution of the given system of equations is:

x = 1 and y = −1

Conclusion

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