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 = a²  …… (1)

bx + ay = b²  …… (2)

Step 1: Write Equations in Standard Form

ax + by − a² = 0  …… (1)

bx + ay − b² = 0  …… (2)

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

From equation (1): a₁ = a, b₁ = b, c₁ = −a²

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

Step 3: Apply Cross-Multiplication Formula

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

Substitute values:

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

x / (a³ − b³) = y / ( −ab(a − b) ) = 1 / (a² − b²)

Step 4: Find the Values of x and y

x / (a³ − b³) = 1 / (a² − b²)

⇒ x = (a² + ab + b²) / (a + b)

y / ( −ab(a − b) ) = 1 / (a² − b²)

⇒ y = −ab / (a + b)

Final Answer

∴ The solution of the given system of equations is:

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

Conclusion

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