Question

The solution of system of equations \(\frac{2}{x} + \frac{3}{y} = 13\) and \(\frac{5}{x} - \frac{4}{y} = -2\) is ______

• A. \(\left(\frac{1}{4}, \frac{1}{3}\right)\)
• B. \(\left(\frac{1}{3}, \frac{1}{4}\right)\)
• C. \(\left(\frac{1}{2}, \frac{1}{3}\right)\)
• D. \(\left(\frac{1}{3}, \frac{1}{2}\right)\)

Answer 1

The Correct Option is C

Step 1: Write the given equations

\( \frac{2}{x} + \frac{3}{y} = 13 \quad \text{(1)} \)
\( \frac{5}{x} - \frac{4}{y} = -2 \quad \text{(2)} \)

Step 2: Introduce substitutions

Let \( \frac{1}{x} = a \) and \( \frac{1}{y} = b \)

Then equations (1) and (2) become:
\( 2a + 3b = 13 \quad \text{(3)} \)
\( 5a - 4b = -2 \quad \text{(4)} \)

Step 3: Eliminate one variable

To eliminate \( b \), multiply equation (3) by 4 and equation (4) by 3:

Equation (3) × 4:
\( 8a + 12b = 52 \quad \text{(5)} \)
Equation (4) × 3:
\( 15a - 12b = -6 \quad \text{(6)} \)

Step 4: Add equations (5) and (6)

\( (8a + 12b) + (15a - 12b) = 52 + (-6) \)
\( 8a + 15a = 23a \)
\( 12b - 12b = 0 \)
So, \( 23a = 46 \Rightarrow a = \frac{46}{23} = 2 \)

Step 5: Substitute the value of \( a \) into equation (3) to find \( b \)

Substitute \( a = 2 \) into \( 2a + 3b = 13 \):
\( 2(2) + 3b = 13 \Rightarrow 4 + 3b = 13 \Rightarrow 3b = 9 \Rightarrow b = \frac{9}{3} = 3 \)

Step 6: Find the values of \( x \) and \( y \)

Since \( a = \frac{1}{x} \), then \( x = \frac{1}{a} = \frac{1}{2} \)
Since \( b = \frac{1}{y} \), then \( y = \frac{1}{b} = \frac{1}{3} \)

The correct option is (C): \(\left(\frac{1}{2}, \frac{1}{3}\right)\)