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:

\[ 2x + y = 35 \\ , 3x + 4y = 65 \]

Solution

Step 1: Compare with Standard Form

The standard form is:

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

From the given equations, we have:

\[ a_1 = 2,\quad b_1 = 1,\quad c_1 = 35 \]

\[ a_2 = 3,\quad b_2 = 4,\quad c_2 = 65 \]

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

\[ \frac{x}{(1 \cdot 65 – 4 \cdot 35)} = \frac{y}{(3 \cdot 35 – 2 \cdot 65)} = \frac{1}{(2 \cdot 4 – 3 \cdot 1)} \]

\[ \frac{x}{(65 – 140)} = \frac{y}{(105 – 130)} = \frac{1}{(8 – 3)} \]

\[ \frac{x}{-75} = \frac{y}{-25} = \frac{1}{5} \]

Step 4: Find the Values of x and y

\[ \frac{x}{-75} = \frac{1}{5} \Rightarrow x = -15 \]

\[ \frac{y}{-25} = \frac{1}{5} \Rightarrow y = -5 \]

Conclusion

The solution of the given system of equations is:

\[ x = -15,\quad y = -5 \]

\[ \therefore \quad \text{The solution is } ( -15,\; -5 ). \]