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  …… (1)

ax + by = a² + b²  …… (2)

Step 1: Convert into Linear Equations

From equation (1):

bx = ay

⇒ bx − ay = 0  …… (1)

Equation (2) is:

ax + by − (a² + b²) = 0  …… (2)

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

From equation (1): a₁ = b, b₁ = −a, c₁ = 0

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

Step 3: Apply Cross-Multiplication Formula

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

Substitute values:

x / [ (−a)(−(a² + b²)) − b(0) ] = y / [ a(0) − b(−(a² + b²)) ] = 1 / [ b(b) − a(−a) ]

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

Step 4: 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).