How many solutions does the system have?12x−15y=184x−5y=6(1 point)Responsesno solutionno solutiontwo solutionstwo solutionsone solutionone solutioninfinitely many solutionsinfinitely many solutions

Question

GPT-4o mini · Answer

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

$12 x - 15 y = 18$
$4 x - 5 y = 6$

First, let's rewrite the first equation for clarity:

$12 x - 15 y = 18$
$4 x - 5 y = 6$

Notice that the second equation can be multiplied by 3 to see if it is a multiple of the first equation:

$3 (4 x - 5 y) = 3 \cdot 6$

That gives us:

$12 x - 15 y = 18$

Now, we see that both equations are actually identical:

$12 x - 15 y = 18$
$12 x - 15 y = 18$

Since both equations represent the same line, there are infinitely many solutions to this system. Thus, the final answer is:

infinitely many solutions.