How many solutions does the system have?

Question

12x−15y=18

4x−5y=6
(1 point)
Responses

two solutions
two solutions

no solution
no solution

infinitely many solutions
infinitely many solutions

one solution

Answer 1

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

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

First, we can simplify equation 2. Notice that if we multiply the entire second equation by 3, we get:

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

Now we see that both equations are actually the same:

1. \( 12x - 15y = 18 \) (original equation 1)
2. \( 12x - 15y = 18 \) (transformed equation 2)

Since both equations represent the same line, the system has infinitely many solutions, as every point on the line is a solution to both equations.

Therefore, the correct response is:

infinitely many solutions.