Question

Which of these systems of equations has no solution?

A. 3a-5b=12
4a-9b=14


B. 3a-5b=14
4a-5b=14


C. 4a-9b=12
4a-9b=12


D. 4a-9b=12
4a-9b=14

Answer 1

To determine which system of equations has no solution, we can analyze each option.

A.
1) \(3a - 5b = 12\)
2) \(4a - 9b = 14\)
This system can potentially have a solution. We can solve it but it's not immediately clear if they are parallel.

B.
1) \(3a - 5b = 14\)
2) \(4a - 5b = 14\)
In this case, we can see that both equations have the same value on the right side, but have different coefficients for \(a\) which means they are not parallel or coincident. We can find unique solutions, hence it has a solution.

C.
1) \(4a - 9b = 12\)
2) \(4a - 9b = 12\)
This system is essentially the same equation repeated, meaning infinitely many solutions exist.

D.
1) \(4a - 9b = 12\)
2) \(4a - 9b = 14\)
Here we have a system where both equations have the same left side but different right sides. This indicates that they are parallel lines which never intersect, meaning there is no solution.

Upon evaluating all of them, the system of equations that has no solution is:

**D.** \(4a - 9b = 12\) and \(4a - 9b = 14\).