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:

\[ ax + by = \frac{a+b}{2} \\ , 3x + 5y = 4 \]

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:

\[ a_1 = a,\quad b_1 = b,\quad c_1 = \frac{a+b}{2} \]

\[ a_2 = 3,\quad b_2 = 5,\quad c_2 = 4 \]

Step 2: Apply Cross-Multiplication Formula

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

Step 3: Substitute the Values

\[ \frac{x}{\left(b\cdot4 – 5\cdot\frac{a+b}{2}\right)} = \frac{y}{\left(a\cdot4 – 3\cdot\frac{a+b}{2}\right)} = \frac{1}{(5a – 3b)} \]

\[ \frac{x}{\frac{5a – 3b}{2}} = \frac{y}{\frac{5a – 3b}{2}} = \frac{1}{(5a – 3b)} \]

Step 4: Find the Values of x and y

\[ x = \frac{1}{2} \]

\[ y = \frac{1}{2} \]

Conclusion

The solution of the given system of equations is:

\[ x = \frac{1}{2},\quad y = \frac{1}{2} \]

\[ \therefore \quad \text{The solution is } \left(\frac{1}{2},\; \frac{1}{2}\right). \]