Question

How many solutions does this system of equations have, given the graphs of the equations? 
3x+2y=8
−6x−4y=8

Answer 1

To determine the number of solutions for the given system of equations, we can analyze the two equations you provided:

1. \( 3x + 2y = 8 \)
2. \( -6x - 4y = 8 \)

First, let's rewrite the second equation in a more manageable form. Notice that you can multiply the first equation by \(-2\) to compare it with the second equation:

Multiplying \( 3x + 2y = 8 \) by \(-2\):

\[
-6x - 4y = -16
\]

Now we have:

1. \( -6x - 4y = -16 \) (from the first equation modified)
2. \( -6x - 4y = 8 \) (the second equation)

Now, we can see that the left sides of both equations are the same:

\[
-6x - 4y = -16
\]
\[
-6x - 4y = 8
\]

Since the left sides are identical, but the right sides are different (−16 ≠ 8), this means that the two equations represent parallel lines.

Parallel lines do not intersect, which means there are no solutions to this system of equations.

Thus, the final answer is:

**The system has no solutions.**