Question

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

4x−5y=6
(1 point)
Responses

infinitely many solutions
infinitely many solutions

one solution
one solution

two solutions
two solutions

no 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 observe that the second equation can be obtained by multiplying the first equation by \( \frac{1}{3} \):

\[
\frac{1}{3}(12x - 15y) = \frac{1}{3}(18)
\]
This simplifies to:
\[
4x - 5y = 6
\]

Since the second equation is essentially the same as the first equation when scaled, both equations represent the same line.

When two equations represent the same line, they have infinitely many solutions, as any point on the line is a solution to both equations.

Thus, the correct response is:

**infinitely many solutions**.