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:

57/(x + y) + 6/(x − y) = 5  …… (1)

38/(x + y) + 21/(x − y) = 9  …… (2)

Step 1: Convert into Linear Equations

Let 1/(x + y) = p and 1/(x − y) = q

Then equation (1) becomes:

57p + 6q = 5  …… (1)

Equation (2) becomes:

38p + 21q = 9  …… (2)

Step 2: Write in Standard Form

57p + 6q − 5 = 0  …… (1)

38p + 21q − 9 = 0  …… (2)

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

From equation (1): a₁ = 57, b₁ = 6, c₁ = −5

From equation (2): a₂ = 38, b₂ = 21, c₂ = −9

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

p / [ 6(−9) − 21(−5) ] = q / [ 38(−5) − 57(−9) ] = 1 / [ 57(21) − 38(6) ]

p / ( −54 + 105 ) = q / ( −190 + 513 ) = 1 / ( 1197 − 228 )

p / 51 = q / 323 = 1 / 969

Step 5: Find the Values of p and q

p / 51 = 1 / 969

⇒ p = 1/19

q / 323 = 1 / 969

⇒ q = 1/3

Step 6: Find the Values of x and y

p = 1/(x + y) = 1/19 ⇒ x + y = 19

q = 1/(x − y) = 1/3 ⇒ x − y = 3

Solving:

2x = 22 ⇒ x = 11

⇒ y = 8

Final Answer

∴ The solution of the given system of equations is:

x = 11 and y = 8

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of linear equations is (11, 8).