Question

The pair of equations \(\frac 32x+\frac 53y=7, \ 9x-10y=12\), represents the following

• A. Parallel lines
• B. No solution
• C. Infinitely many solutions
• D. One solution

Answer 1

The Correct Option is D

To solve the problem, we need to analyze the pair of linear equations and determine their geometric relationship or the number of solutions they have.

1. Given Equations:
Equation (1): \( \frac{3}{2}x + \frac{5}{3}y = 7 \) 
Equation (2): \( 9x - 10y = 12 \)

2. Convert Both Equations to Standard Form:
Multiply equation (1) by 6 to eliminate denominators:
\( 6 \left( \frac{3}{2}x + \frac{5}{3}y \right) = 6 \cdot 7 \)
⇒ \( 9x + 10y = 42 \)

Now, the system becomes:
Equation (1): \( 9x + 10y = 42 \) 
Equation (2): \( 9x - 10y = 12 \)

3. Solve the System of Equations:
Add the two equations:
\( (9x + 10y) + (9x - 10y) = 42 + 12 \)
⇒ \( 18x = 54 \)
⇒ \( x = 3 \)

Now substitute \( x = 3 \) in Equation (1):
\( 9(3) + 10y = 42 \)
⇒ \( 27 + 10y = 42 \)
⇒ \( 10y = 15 \)
⇒ \( y = 1.5 \)

4. Conclusion:
The pair of equations has a unique solution: \( x = 3, y = 1.5 \).
Therefore, the lines intersect at a single point.

Final Answer:
The system of equations has one solution.