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:

\[ \frac{b}{a}x + \frac{a}{b}y = a^2 + b^2, \\ x + y = 2ab \]

Solution

Step 1: Convert the First Equation into Linear Form

Multiply the first equation by \(ab\):

\[ b^2x + a^2y = ab(a^2 + b^2) \quad \text{(1)} \]

The second equation is:

\[ x + y = 2ab \quad \text{(2)} \]

Step 2: Compare with the Standard Form

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

From (1) and (2), we get:

\[ a_1 = b^2,\quad b_1 = a^2,\quad c_1 = ab(a^2+b^2) \]

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

Step 3: Apply Cross-Multiplication Formula

\[ \frac{x}{(b_1c_2 – b_2c_1)} = \frac{y}{(a_2c_1 – a_1c_2)} = \frac{1}{(a_1b_2 – a_2b_1)} \]

Step 4: Substitute the Values

\[ \frac{x}{\big[a^2(2ab) – 1\cdot ab(a^2+b^2)\big]} = \frac{y}{\big[1\cdot ab(a^2+b^2) – b^2(2ab)\big]} = \frac{1}{(b^2 – a^2)} \]

\[ \frac{x}{ab(a^2-b^2)} = \frac{y}{ab(b^2-a^2)} = \frac{1}{(b^2-a^2)} \]

Step 5: Find the Values of x and y

\[ x = ab \]

\[ y = ab \]

Conclusion

The solution of the given system of equations is:

\[ x = ab,\quad y = ab \]

\[ \therefore \quad \text{The solution is } (ab,\; ab). \]