To determine how many solutions the system of equations has, we can analyze the two equations given:
- \( 12x - 15y = 18 \)
- \( 4x - 5y = 6 \)
First, let's rewrite the second equation to see if it can be aligned with the first. We can multiply the second equation by 3 to make it easier to compare:
\[ 3(4x - 5y) = 3(6) \] \[ 12x - 15y = 18 \]
Now we see that both equations are identical:
- \( 12x - 15y = 18 \)
- \( 12x - 15y = 18 \)
Since both equations are the same, this indicates that there are infinitely many solutions to the system. The solutions can be expressed as a line in the coordinate plane, where any point on that line is a solution to the system.
The correct response is:
infinitely many solutions