Question

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Solving with Elimination Quick Check
5 of 55 of 5 Items








Question
How many solutions does the system have?
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 relationships between the two equations:

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

First, we can try to express the second equation in a similar form as the first one. Notice that if we multiply the second equation by 3, we get:

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

This means that the two equations are actually the same equation. Therefore, they represent the same line in the coordinate plane.

Since the two equations are dependent and represent the same line, the system has **infinitely many solutions**.

So, the correct response is:

**infinitely many solutions.**