Question

Question
How many solutions does the system have?
12x−15y=18

4x−5y=6
(1 point)
Responses

two solutions
two solutions

one solution
one solution

no solution
no solution

infinitely many solutions

Answer 1

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

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

Let's see if one equation is a multiple of the other.

We can simplify the second equation:

Multiply the second equation by 3:

\[
3(4x - 5y) = 3(6) \implies 12x - 15y = 18
\]

Now, we can see that the first equation is indeed the same as the transformed second equation. Thus, we have:

\[
12x - 15y = 18 \quad \text{(first equation)}
\]
\[
12x - 15y = 18 \quad \text{(transformed second equation)}
\]

Since both equations represent the same line, this means there are infinitely many solutions.

The correct response is:

**infinitely many solutions**