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 + ay = b  …… (1)

ax − by = c  …… (2)

Step 1: Write Equations in Standard Form

x + ay − b = 0  …… (1)

ax − by − c = 0  …… (2)

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

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

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

Step 3: Apply Cross-Multiplication Formula

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

Substitute values:

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

x / ( −ac − b² ) = y / ( −ab + c ) = 1 / ( −b − a² )

Step 4: Find the Values of x and y

x / ( −ac − b² ) = 1 / ( −b − a² )

⇒ x = (ac + b²) / (a² + b)

y / ( −ab + c ) = 1 / ( −b − a² )

⇒ y = (ab − c) / (a² + b)

Final Answer

∴ The solution of the given system of equations is:

x = (ac + b²) / (a² + b)
y = (ab − c) / (a² + b)

Conclusion

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