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:

2ax + 3by = a + 2b  …… (1)

3ax + 2by = 2a + b  …… (2)

Step 1: Write Equations in Standard Form

2ax + 3by − (a + 2b) = 0  …… (1)

3ax + 2by − (2a + b) = 0  …… (2)

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

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

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

Step 3: Apply Cross-Multiplication Formula

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

Substitute values:

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

x / ( −6ab − 3b² + 2ab + 4b² ) = y / ( −3a² − 6ab + 4a² + 2ab ) = 1 / ( 4ab − 9ab )

x / ( b² − 4ab ) = y / ( a² − 4ab ) = 1 / ( −5ab )

Step 4: Find the Values of x and y

x / ( b² − 4ab ) = 1 / ( −5ab )

⇒ x = (4ab − b²) / (5ab)

⇒ x = (4a − b) / (5a)

y / ( a² − 4ab ) = 1 / ( −5ab )

⇒ y = (4ab − a²) / (5ab)

⇒ y = (4b − a) / (5b)

Final Answer

∴ The solution of the given system of equations is:

x = (4a − b) / (5a)
y = (4b − a) / (5b)

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of linear equations is ( (4a − b)/(5a), (4b − a)/(5b) ).