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:

mx − ny = m² + n²  …… (1)

x + y = 2m  …… (2)

Step 1: Write Equations in Standard Form

mx − ny − (m² + n²) = 0  …… (1)

x + y − 2m = 0  …… (2)

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

From equation (1): a₁ = m, b₁ = −n, c₁ = −(m² + n²)

From equation (2): a₂ = 1, b₂ = 1, c₂ = −2m

Step 3: Apply Cross-Multiplication Formula

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

Substitute values:

x / [ (−n)(−2m) − 1(−(m² + n²)) ] = y / [ 1(−(m² + n²)) − m(−2m) ] = 1 / [ m(1) − 1(−n) ]

x / (2mn + m² + n²) = y / (−m² − n² + 2m²) = 1 / (m + n)

x / (m + n)² = y / (m² − n²) = 1 / (m + n)

Step 4: Find the Values of x and y

x / (m + n)² = 1 / (m + n)

⇒ x = m + n

y / (m² − n²) = 1 / (m + n)

⇒ y = (m² − n²) / (m + n)

⇒ y = m − n

Final Answer

∴ The solution of the given system of equations is:

x = m + n and y = m − n

Conclusion

Thus, by using the cross-multiplication method, we find that the solution of the given system of linear equations is (m + n, m − n).