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 + y) / xy = 2  …… (1)

(x − y) / xy = 6  …… (2)

Step 1: Convert into Linear Equations

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

Then equation (1) becomes:

p + q = 2  …… (1)

Equation (2) becomes:

p − q = 6  …… (2)

Step 2: Write in Standard Form

p + q − 2 = 0  …… (1)

p − q − 6 = 0  …… (2)

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

From equation (1): a₁ = 1, b₁ = 1, c₁ = −2

From equation (2): a₂ = 1, b₂ = −1, c₂ = −6

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

p / [1(−6) − (−1)(−2)] = q / [1(−2) − 1(−6)] = 1 / [1(−1) − 1(1)]

p / (−6 − 2) = q / (−2 + 6) = 1 / (−1 − 1)

p / (−8) = q / (4) = 1 / (−2)

Step 5: Find the Values of p and q

p / (−8) = 1 / (−2)

⇒ p = 4

q / 4 = 1 / (−2)

⇒ q = −2

Step 6: Find the Values of x and y

p = 1/x = 4 ⇒ x = 1/4

q = 1/y = −2 ⇒ y = −1/2

Final Answer

∴ The solution of the given system of equations is:

x = 1/4 and y = −1/2

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of equations is (1/4, −1/2).