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:

(a − b)x + (a + b)y = 2a² − 2b²  …… (1)

(a + b)(x + y) = 4ab  …… (2)

Step 1: Simplify Equation (2)

Expand equation (2):

(a + b)x + (a + b)y = 4ab  …… (2)

Step 2: Write Equations in Standard Form

(a − b)x + (a + b)y − (2a² − 2b²) = 0  …… (1)

(a + b)x + (a + b)y − 4ab = 0  …… (2)

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

From equation (1): a₁ = (a − b), b₁ = (a + b), c₁ = −(2a² − 2b²)

From equation (2): a₂ = (a + b), b₂ = (a + b), c₂ = −4ab

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

x / [ (a + b)(−4ab) − (a + b)(−(2a² − 2b²)) ] = y / [ (a + b)(−(2a² − 2b²)) − (a − b)(−4ab) ] = 1 / [ (a − b)(a + b) − (a + b)(a + b) ]

x / [ (a + b)(2a² − 2b² − 4ab) ] = y / [ −2(a + b)(a² − b²) + 4ab(a − b) ] = 1 / [ −2b(a + b) ]

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

Step 5: Find the Values of x and y

x / [ 2(a + b)(a − b)² ] = 1 / [ −2b(a + b) ]

⇒ x = (b − a)/b

y / [ −2(a − b)(a + b)² ] = 1 / [ −2b(a + b) ]

⇒ y = (a − b)/a

Final Answer

∴ The solution of the given system of equations is:

x = (b − a)/b
y = (a − b)/a

Conclusion

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