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:

\[ \frac{ax}{b} – \frac{by}{a} = a + b, \\ ax – by = 2ab \]

Solution

Step 1: Convert the First Equation into Linear Form

Multiply the first equation by \(ab\):

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

The second equation is:

\[ ax – by = 2ab \quad \text{(2)} \]

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 have:

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

\[ a_2 = a,\quad b_2 = -b,\quad c_2 = 2ab \]

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

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

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

Step 5: Find the Values of x and y

\[ x = -b \]

\[ y = a \]

Conclusion

The solution of the given system of equations is:

\[ x = -b,\quad y = a \]

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