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

To determine whether the pair of linear equations is consistent or inconsistent, we need to analyze their forms and compare their coefficients.

Given equations are:

• \(\frac{3x}{2} + \frac{5y}{3} = 7\)
• \(9x + 10y = 14\)

First, we will work on simplifying the first equation:

Convert the first equation into standard form:

Multiply through by 6 (the LCM of 2 and 3) to clear the fractions:

• \(6 \left(\frac{3x}{2} + \frac{5y}{3}\right) = 6 \cdot 7\)
• \(9x + 10y = 42\)

Now, compare both equations:

• First equation (equivalent form): \(9x + 10y = 42\)
• Second equation: \(9x + 10y = 14\)

Both equations have the same left-hand side but different right-hand sides. This implies:

• The lines represented by these equations are parallel.
• Since the lines are parallel, they do not intersect.

The absence of an intersection point makes the system inconsistent because there is no common solution.

Hence, the correct option is: inconsistent.