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:
\[ a^2x + b^2y = c^2 \\ , b^2x + a^2y = a^2 \]
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:
\[ a_1 = a^2,\quad b_1 = b^2,\quad c_1 = c^2 \]
\[ a_2 = b^2,\quad b_2 = a^2,\quad c_2 = a^2 \]
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
\[ \frac{x}{(b^2\cdot a^2 – a^2\cdot c^2)} = \frac{y}{(b^2\cdot c^2 – a^2\cdot a^2)} = \frac{1}{(a^2\cdot a^2 – b^2\cdot b^2)} \]
\[ \frac{x}{a^2(b^2 – c^2)} = \frac{y}{(b^2c^2 – a^4)} = \frac{1}{(a^4 – b^4)} \]
Step 4: Find the Values of x and y
\[ x = \frac{b^2 – c^2}{a^2 – b^2} \]
\[ y = \frac{a^2 – c^2}{a^2 – b^2} \]
Conclusion
The solution of the given system of equations is:
\[ x = \frac{b^2 – c^2}{a^2 – b^2},\quad y = \frac{a^2 – c^2}{a^2 – b^2} \]
\[ \therefore \quad \text{The solution is } \left( \frac{b^2 – c^2}{a^2 – b^2}, \; \frac{a^2 – c^2}{a^2 – b^2} \right). \]