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:

\[ 2ax + 3by = a + 2b \\ , 3ax + 2by = 2a + b \]

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 have:

\[ a_1 = 2a,\quad b_1 = 3b,\quad c_1 = a + 2b \]

\[ a_2 = 3a,\quad b_2 = 2b,\quad c_2 = 2a + b \]

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

\[ \frac{x}{\big[3b(2a+b) – 2b(a+2b)\big]} = \frac{y}{\big[3a(a+2b) – 2a(2a+b)\big]} = \frac{1}{\big[2a(2b) – 3a(3b)\big]} \]

\[ \frac{x}{b(b-4a)} = \frac{y}{a(a-4b)} = \frac{1}{-5ab} \]

Step 4: Find the Values of x and y

\[ x = \frac{4a – b}{5a} \]

\[ y = \frac{a – 4b}{5b} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{4a – b}{5a},\quad y = \frac{a – 4b}{5b} \]

\[ \therefore \quad \text{The solution is } \left( \frac{4a – b}{5a}, \; \frac{a – 4b}{5b} \right). \]