Question

Question
How many solutions does the system of equations have?
8x−6y=48

8x+6y=48
(1 point)
Responses

two
two

none
none

one
one

infinite

Answer 1

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

1. \( 8x - 6y = 48 \)
2. \( 8x + 6y = 48 \)

First, let's rewrite both equations:

From the first equation:
\[ 8x - 6y = 48 \]
\[ 6y = 8x - 48 \]
\[ y = \frac{8}{6}x - 8 = \frac{4}{3}x - 8 \]

From the second equation:
\[ 8x + 6y = 48 \]
\[ 6y = 48 - 8x \]
\[ y = \frac{48 - 8x}{6} = 8 - \frac{4}{3}x \]

Now we can see the slope-intercept forms of both equations:
1. \( y = \frac{4}{3}x - 8 \)
2. \( y = -\frac{4}{3}x + 8 \)

The slopes are different (\(\frac{4}{3}\) and \(-\frac{4}{3}\)), which indicates that the lines represented by these equations are not parallel and will intersect at exactly one point.

Therefore, the system of equations has exactly **one solution**.

The answer is: **one**.