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 = a + b  …… (1)

x/a² + y/b² = 2  …… (2)

Step 1: Convert into Linear Equations

Let x/a = p and y/b = q

Then equation (1) becomes:

p + q = a + b  …… (1)

Equation (2) becomes:

p/a + q/b = 2  …… (2)

Step 2: Write in Standard Form

p + q − (a + b) = 0  …… (1)

p/a + q/b − 2 = 0  …… (2)

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

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

From equation (2): a₂ = 1/a, b₂ = 1/b, c₂ = −2

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

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

p / [ −2 + (a + b)/b ] = q / [ −(a + b)/a + 2 ] = 1 / [ (a − b)/(ab) ]

Step 5: Find the Values of p and q

On simplifying, we get:

p = a

q = b

Step 6: Find the Values of x and y

p = x/a = a ⇒ x = a²

q = y/b = b ⇒ 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).