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:

\[ x + ay = b \\ , ax – by = c \]

Solution

Step 1: Compare with the Standard Form

The standard form is:

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

From the given equations, we get:

\[ a_1 = 1,\quad b_1 = a,\quad c_1 = b \]

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

Step 2: 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 3: Substitute the Values

\[ \frac{x}{(a\cdot c – (-b)\cdot b)} = \frac{y}{(a\cdot b – 1\cdot c)} = \frac{1}{(1\cdot (-b) – a\cdot a)} \]

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

Step 4: Find the Values of x and y

\[ \frac{x}{(ac + b^2)} = \frac{1}{-(a^2 + b)} \Rightarrow x = -\frac{ac + b^2}{a^2 + b} \]

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

Conclusion

The solution of the given system of equations is:

\[ x = -\frac{ac + b^2}{a^2 + b},\quad y = \frac{c – ab}{a^2 + b} \]

\[ \therefore \quad \text{The solution is } \left( -\frac{ac + b^2}{a^2 + b}, \; \frac{c – ab}{a^2 + b} \right). \]