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/b − by/a = a + b  …… (1)

ax − by = 2ab  …… (2)

Step 1: Convert Equation (1) into Linear Form

Multiply equation (1) by ab:

a²x − b²y = ab(a + b)  …… (1)

Step 2: Write Equations in Standard Form

a²x − b²y − ab(a + b) = 0  …… (1)

ax − by − 2ab = 0  …… (2)

Step 3: Compare with ax + by + c = 0

From equation (1): a₁ = a², b₁ = −b², c₁ = −ab(a + b)

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

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

x / [ (−b²)(−2ab) − (−b)(−ab(a + b)) ] = y / [ a(−ab(a + b)) − a²(−2ab) ] = 1 / [ a²(−b) − a(−b²) ]

x / [ 2ab³ − ab²(a + b) ] = y / [ −a²b(a + b) + 2a³b ] = 1 / [ ab(b − a) ]

x / [ ab²(b − a) ] = y / [ a²b(a − b) ] = 1 / [ ab(b − a) ]

Step 5: Find the Values of x and y

x / [ ab²(b − a) ] = 1 / [ ab(b − a) ]

⇒ x = b

y / [ a²b(a − b) ] = 1 / [ ab(b − a) ]

⇒ y = −a

Final Answer

∴ The solution of the given system of equations is:

x = b and y = −a

Conclusion

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