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:

(a + 2b)x + (2a − b)y = 2  …… (1)

(a − 2b)x + (2a + b)y = 3  …… (2)

Step 1: Write Equations in Standard Form

(a + 2b)x + (2a − b)y − 2 = 0  …… (1)

(a − 2b)x + (2a + b)y − 3 = 0  …… (2)

Step 2: Compare with ax + by + c = 0

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

From equation (2): a₂ = (a − 2b), b₂ = (2a + b), c₂ = −3

Step 3: Apply Cross-Multiplication Formula

x / (b₁c₂ − b₂c₁) = y / (a₂c₁ − a₁c₂) = 1 / (a₁b₂ − a₂b₁)

Substitute values:

x / [ (2a − b)(−3) − (2a + b)(−2) ] = y / [ (a − 2b)(−2) − (a + 2b)(−3) ] = 1 / [ (a + 2b)(2a + b) − (a − 2b)(2a − b) ]

x / ( −6a + 3b + 4a + 2b ) = y / ( −2a + 4b + 3a + 6b ) = 1 / ( 10ab )

x / ( −2a + 5b ) = y / ( a + 10b ) = 1 / ( 10ab )

Step 4: Find the Values of x and y

x / ( −2a + 5b ) = 1 / ( 10ab )

⇒ x = (5b − 2a) / (10ab)

y / ( a + 10b ) = 1 / ( 10ab )

⇒ y = (a + 10b) / (10ab)

Final Answer

∴ The solution of the given system of equations is:

x = (5b − 2a) / (10ab)
y = (a + 10b) / (10ab)

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of linear equations is ( (5b − 2a)/(10ab), (a + 10b)/(10ab) ).