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:

5/(x + y) − 2/(x − y) = −1  …… (1)

15/(x + y) + 7/(x − y) = 10  …… (2)

where x ≠ 0 and y ≠ 0

Step 1: Convert into Linear Equations

Let 1/(x + y) = p and 1/(x − y) = q

Then equation (1) becomes:

5p − 2q = −1  …… (1)

Equation (2) becomes:

15p + 7q = 10  …… (2)

Step 2: Write in Standard Form

5p − 2q + 1 = 0  …… (1)

15p + 7q − 10 = 0  …… (2)

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

From equation (1): a₁ = 5, b₁ = −2, c₁ = 1

From equation (2): a₂ = 15, b₂ = 7, c₂ = −10

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

p / [ (−2)(−10) − 7(1) ] = q / [ 15(1) − 5(−10) ] = 1 / [ 5(7) − 15(−2) ]

p / (20 − 7) = q / (15 + 50) = 1 / (35 + 30)

p / 13 = q / 65 = 1 / 65

Step 5: Find the Values of p and q

p / 13 = 1 / 65

⇒ p = 1/5

q / 65 = 1 / 65

⇒ q = 1

Step 6: Find the Values of x and y

p = 1/(x + y) = 1/5 ⇒ x + y = 5

q = 1/(x − y) = 1 ⇒ x − y = 1

Solving:

x + y = 5

x − y = 1

⇒ 2x = 6 ⇒ x = 3

⇒ y = 2

Final Answer

∴ The solution of the given system of equations is:

x = 3 and y = 2

Conclusion

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