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 no solution no solution one solution

Answer 1

To determine how many solutions the system of equations has, we can analyze the given equations:

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

First, let's simplify both equations.

Starting with the second equation:

\[ 4x - 5y = 6 \]

We can multiply the entire equation by 3 to help us compare it to the first equation:

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

Now we can see that the second equation \( 4x - 5y = 6 \) is simply a scaled version of the first equation \( 12x - 15y = 18 \).

Since both equations represent the same line, they have infinitely many solutions; every point on the line \( 12x - 15y = 18 \) (or equivalently \( 4x - 5y = 6 \)) is a solution to the system.

Thus, the answer is infinitely many solutions.