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:

2/x + 3/y = 13  …… (1)

5/x − 4/y = −2  …… (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:

2p + 3q = 13  …… (1)

Equation (2) becomes:

5p − 4q = −2  …… (2)

Step 2: Write in Standard Form

2p + 3q − 13 = 0  …… (1)

5p − 4q + 2 = 0  …… (2)

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

From equation (1): a₁ = 2, b₁ = 3, c₁ = −13

From equation (2): a₂ = 5, b₂ = −4, c₂ = 2

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

p / [ 3(2) − (−4)(−13) ] = q / [ 5(−13) − 2(2) ] = 1 / [ 2(−4) − 5(3) ]

p / (6 − 52) = q / (−65 − 4) = 1 / (−8 − 15)

p / (−46) = q / (−69) = 1 / (−23)

Step 5: Find the Values of p and q

p / (−46) = 1 / (−23)

⇒ p = 2

q / (−69) = 1 / (−23)

⇒ q = 3

Step 6: Find the Values of x and y

p = 1/x = 2 ⇒ x = 1/2

q = 1/y = 3 ⇒ y = 1/3

Final Answer

∴ The solution of the given system of equations is:

x = 1/2 and y = 1/3

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of linear equations is (1/2, 1/3).