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:

\[ 2x – y = 6 \\ , x – y = 2 \]

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 have:

\[ a_1 = 2,\quad b_1 = -1,\quad c_1 = 6 \]

\[ a_2 = 1,\quad b_2 = -1,\quad c_2 = 2 \]

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

\[ \frac{x}{((-1)\cdot 2 – (-1)\cdot 6)} = \frac{y}{(1\cdot 6 – 2\cdot 2)} = \frac{1}{(2\cdot (-1) – 1\cdot (-1))} \]

\[ \frac{x}{(-2 + 6)} = \frac{y}{(6 – 4)} = \frac{1}{(-2 + 1)} \]

\[ \frac{x}{4} = \frac{y}{2} = \frac{1}{-1} \]

Step 4: Find the Values of x and y

\[ \frac{x}{4} = \frac{1}{-1} \Rightarrow x = -4 \]

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

Conclusion

The solution of the given system of equations is:

\[ x = 4,\quad y = 2 \]

\[ \therefore \quad \text{The solution is } ( 4,\; 2 ). \]