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:

ax + by = (a + b)/2  …… (1)

3x + 5y = 4  …… (2)

Step 1: Write Equations in Standard Form

2ax + 2by − (a + b) = 0  …… (1)

3x + 5y − 4 = 0  …… (2)

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

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

From equation (2): a₂ = 3, b₂ = 5, c₂ = −4

Step 3: Apply Cross-Multiplication Formula

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

Substitute values:

x / [ 2b(−4) − 5(−(a + b)) ] = y / [ 3(−(a + b)) − 2a(−4) ] = 1 / [ 2a(5) − 3(2b) ]

x / ( −8b + 5a + 5b ) = y / ( −3a − 3b + 8a ) = 1 / ( 10a − 6b )

x / (5a − 3b) = y / (5a − 3b) = 1 / (10a − 6b)

Step 4: Find the Values of x and y

x / (5a − 3b) = 1 / (10a − 6b)

⇒ x = 1/2

y / (5a − 3b) = 1 / (10a − 6b)

⇒ y = 1/2

Final Answer

∴ The solution of the given system of equations is:

x = 1/2 and y = 1/2

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of linear equations is (1/2, 1/2).