Solve the System of Equations by the Method of Cross-Multiplication

Video Explanation

Question

Solve the following system of equations by the method of cross-multiplication:

\[ mx – ny = m^2 + n^2 \\ x + y = 2m \]

Solution

Step 1: Compare with the Standard Form

\[ a_1x + b_1y = c_1,\quad a_2x + b_2y = c_2 \]

From the given equations:

\[ a_1 = m,\quad b_1 = -n,\quad c_1 = m^2 + n^2 \]

\[ a_2 = 1,\quad b_2 = 1,\quad c_2 = 2m \]

Step 2: Apply Cross-Multiplication Formula

\[ \frac{x}{(c_1b_2 – c_2b_1)} = \frac{y}{(a_1c_2 – a_2c_1)} = \frac{1}{(a_1b_2 – a_2b_1)} \]

Step 3: Substitute the Values

\[ \frac{x}{\big[(m^2+n^2)(1) – (2m)(-n)\big]} = \frac{y}{\big[m(2m) – 1(m^2+n^2)\big]} = \frac{1}{\big[m(1) – 1(-n)\big]} \]

\[ \frac{x}{m^2+n^2+2mn} = \frac{y}{m^2-n^2} = \frac{1}{m+n} \]

Step 4: Find the Values of x and y

\[ x = \frac{(m+n)^2}{m+n} = m+n \]

\[ y = \frac{(m-n)(m+n)}{m+n} = m-n \]

Conclusion

The solution of the given system of equations is:

\[ x = m+n,\quad y = m-n \]

\[ \therefore \quad \text{The solution is } (m+n,\; m-n). \]