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:

6(ax + by) = 3a + 2b  …… (1)

6(bx − ay) = 3b − 2a  …… (2)

Step 1: Simplify the Given Equations

From equation (1):

ax + by = (3a + 2b)/6

⇒ 6ax + 6by − 3a − 2b = 0  …… (1)

From equation (2):

bx − ay = (3b − 2a)/6

⇒ 6bx − 6ay − 3b + 2a = 0  …… (2)

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

From equation (1): a₁ = 6a, b₁ = 6b, c₁ = −(3a + 2b)

From equation (2): a₂ = 6b, b₂ = −6a, c₂ = −(3b − 2a)

Step 3: Apply Cross-Multiplication Formula

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

Substitute values:

x / [ 6b(−(3b − 2a)) − (−6a)(−(3a + 2b)) ] = y / [ 6b(−(3a + 2b)) − 6a(−(3b − 2a)) ] = 1 / [ 6a(−6a) − 6b(6b) ]

x / [ −18b² + 12ab − 18a² − 12ab ] = y / [ −18ab − 12b² + 18ab − 12a² ] = 1 / [ −36(a² + b²) ]

x / [ −18(a² + b²) ] = y / [ −12(a² + b²) ] = 1 / [ −36(a² + b²) ]

Step 4: Find the Values of x and y

x / [ −18(a² + b²) ] = 1 / [ −36(a² + b²) ]

⇒ x = 1/2

y / [ −12(a² + b²) ] = 1 / [ −36(a² + b²) ]

⇒ 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).