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:

bx + cy = a + b  …… (1)

ax{1/(a − b) − 1/(a + b)} + cy{1/(b − a) − 1/(b + a)} = 2a/(a + b)  …… (2)

Step 1: Simplify Equation (2)

1/(a − b) − 1/(a + b) = (2b)/(a² − b²)

1/(b − a) − 1/(b + a) = (−2a)/(a² − b²)

Substitute in equation (2):

ax · (2b)/(a² − b²) − cy · (2a)/(a² − b²) = 2a/(a + b)

Multiply both sides by (a² − b²):

2abx − 2acy = 2a(a − b)

⇒ bx − cy = a − b  …… (2)

Step 2: Write Equations in Standard Form

bx + cy − (a + b) = 0  …… (1)

bx − cy − (a − b) = 0  …… (2)

Step 3: Compare with ax + by + c = 0

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

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

Step 4: Apply Cross-Multiplication Formula

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

Substitute values:

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

x / ( −c(a − b) − c(a + b) ) = y / ( −b(a + b) + b(a − b) ) = 1 / ( −2bc )

x / ( −2ac ) = y / ( −2b² ) = 1 / ( −2bc )

Step 5: Find the Values of x and y

x / ( −2ac ) = 1 / ( −2bc )

⇒ x = a/b

y / ( −2b² ) = 1 / ( −2bc )

⇒ y = b/c

Final Answer

∴ The solution of the given system of equations is:

x = a/b and y = b/c

Conclusion

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