Question

How many solutions does the system have?
12x−15y=18
4x−5y=6 (1 point)
Responses

infinitely many solutions
infinitely many solutions

two solutions
two solutions

one solution
one solution

no soluti

Answer 1

To determine the number of solutions for the given system of equations, we can analyze the equations:

1. \( 12x - 15y = 18 \)
2. \( 4x - 5y = 6 \)

First, we can see if the second equation can be derived from the first one. Notice that the second equation can be transformed to look similar to the first equation:

If we multiply the second equation by 3:

\[
3(4x - 5y) = 3(6)
\]
This simplifies to:

\[
12x - 15y = 18
\]

Now we see that both equations are actually the same:

\[
12x - 15y = 18
\]

Since both equations are identical, they represent the same line. Therefore, there are infinitely many solutions to this system because every point on the line represented by either equation is a solution.

**Conclusion:** The answer is **infinitely many solutions**.