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:

a²/x − b²/y = 0  …… (1)

a²b/x + b²a/y = a + b  …… (2)

where x ≠ 0 and y ≠ 0

Step 1: Convert into Linear Equations

Let 1/x = p and 1/y = q

Then equation (1) becomes:

a²p − b²q = 0  …… (1)

Equation (2) becomes:

a²b p + b²a q = a + b  …… (2)

Step 2: Write in Standard Form

a²p − b²q = 0  …… (1)

a²b p + b²a q − (a + b) = 0  …… (2)

Step 3: Compare with ap + bq + c = 0

From equation (1): a₁ = a², b₁ = −b², c₁ = 0

From equation (2): a₂ = a²b, b₂ = b²a, c₂ = −(a + b)

Step 4: Apply Cross-Multiplication Formula

p / (b₁c₂ − b₂c₁) = q / (a₂c₁ − a₁c₂) = 1 / (a₁b₂ − a₂b₁)

Substitute values:

p / [ (−b²)(−(a + b)) − (ab²)(0) ] = q / [ (a²b)(0) − a²(−(a + b)) ] = 1 / [ a²(ab²) − (a²b)(−b²) ]

p / [ b²(a + b) ] = q / [ a²(a + b) ] = 1 / [ 2a²b² ]

Step 5: Find the Values of p and q

p = 1 / a²

q = 1 / b²

Step 6: Find the Values of x and y

p = 1/x = 1/a² ⇒ x = a²

q = 1/y = 1/b² ⇒ y = b²

Final Answer

∴ The solution of the given system of equations is:

x = a² and y = b²

Conclusion

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