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:

a²x + b²y = c²  …… (1)

b²x + a²y = a²  …… (2)

Step 1: Write Equations in Standard Form

a²x + b²y − c² = 0  …… (1)

b²x + a²y − a² = 0  …… (2)

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

From equation (1): a₁ = a², b₁ = b², c₁ = −c²

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

Step 3: Apply Cross-Multiplication Formula

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

Substitute values:

x / [ b²(−a²) − a²(−c²) ] = y / [ b²(−c²) − a²(−a²) ] = 1 / [ a²a² − b²b² ]

x / [ a²(c² − b²) ] = y / [ a⁴ − b²c² ] = 1 / [ a⁴ − b⁴ ]

Step 4: Find the Values of x and y

x = a²(c² − b²) / (a⁴ − b⁴)

y = (a⁴ − b²c²) / (a⁴ − b⁴)

Final Answer

∴ The solution of the given system of equations is:

x = a²(c² − b²) / (a⁴ − b⁴)
y = (a⁴ − b²c²) / (a⁴ − b⁴)

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of linear equations is ( a²(c² − b²) / (a⁴ − b⁴), (a⁴ − b²c²) / (a⁴ − b⁴) ).