Question

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

4x−5y=6
(1 point)
Responses

one solution
one solution

no solution
no solution

infinitely many solutions
infinitely many solutions

two solutions

Answer 1

To determine the number of solutions for the system of equations given by \(12x - 15y = 18\) and \(4x - 5y = 6\), we can start by rewriting both equations in a simpler form.

First, let's simplify the second equation:

\[
4x - 5y = 6
\]

Now, multiply both sides of this equation by 3 to align it with the coefficients of the first equation:

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

Now we can see that both equations:

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

are exactly the same.

When two equations are the same, they represent the same line on a graph. Therefore, there are infinitely many solutions to this system, as every point on the line (each \((x, y)\) pair satisfying the equation) is a solution.

So the final answer is:

**infinitely many solutions**.