Question

The pair of linear equations \( \frac{3x}{2} + \frac{5y}{3} = 7 \) and \( 9x + 10y = 14 \), is :

• A. consistent
• B. inconsistent
• C. consistent with one solution
• D. consistent with many solutions

Hint:

Always convert fractional equations to integer form first. If the left sides are identical but the constants on the right are different, the lines are parallel and the system is inconsistent.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A system of linear equations \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \) is inconsistent if the lines are parallel and never intersect. This happens when:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]
Step 2: Key Formula or Approach:
Simplify the fractional coefficients of the first equation to compare the ratios effectively.
Step 3: Detailed Explanation:
First equation: \( \frac{3}{2}x + \frac{5}{3}y = 7 \)
Multiply the entire equation by the LCM of denominators (6):
\[ 6 \left( \frac{3}{2}x \right) + 6 \left( \frac{5}{3}y \right) = 6(7) \]
\[ 9x + 10y = 42 \]
Comparing this with the second equation: \( 9x + 10y = 14 \).
Identify coefficients:
\( a_1 = 9, b_1 = 10, c_1 = 42 \)
\( a_2 = 9, b_2 = 10, c_2 = 14 \)
Calculate ratios:
\[ \frac{a_1}{a_2} = \frac{9}{9} = 1 \]
\[ \frac{b_1}{b_2} = \frac{10}{10} = 1 \]
\[ \frac{c_1}{c_2} = \frac{42}{14} = 3 \]
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the equations represent parallel lines.
Step 4: Final Answer:
The pair of equations is inconsistent.