Question

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

• A. Consistent
• B. Inconsistent
• C. Dependent
• D. None of these

Hint:

To determine if a system of equations is consistent, check if the equations have a common solution. If no solution exists, the system is inconsistent.

Answer 1

The Correct Option is B

To check the consistency of the system of equations, we first convert both equations to a common format. The first equation is: \[ \frac{3}{2}x + \frac{5}{3}y = 7 \quad \Rightarrow \quad 9x + 10y = 42 \quad \text{(multiplying both sides by 6)}. \] The second equation is: \[ 9x - 10y = 14. \] Now, we have the system: \[ 9x + 10y = 42 \quad \text{(1)}, \] \[ 9x - 10y = 14 \quad \text{(2)}. \] Adding equations (1) and (2): \[ (9x + 10y) + (9x - 10y) = 42 + 14 \quad \Rightarrow \quad 18x = 56 \quad \Rightarrow \quad x = \frac{56}{18} = \frac{28}{9}. \] Substitute \( x = \frac{28}{9} \) into one of the equations (e.g., equation 2): \[ 9x - 10y = 14 \quad \Rightarrow \quad 9 \times \frac{28}{9} - 10y = 14 \quad \Rightarrow \quad 28 - 10y = 14 \quad \Rightarrow \quad 10y = 14 \quad \Rightarrow \quad y = 1. \] Thus, the system is inconsistent and has no solution. The system is \( \boxed{\text{Inconsistent}} \).