Question

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Solving with Elimination Quick Check
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Question
How many solutions does the system of equations have?
8x−6y=48

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

none
none

infinite
infinite

two
two

one
one

Answer 1

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

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

We can rearrange both equations for easier comparison.

From the first equation, we can express \(y\):
\[ 8x - 6y = 48 \]
\[ -6y = 48 - 8x \]
\[ y = \frac{8x - 48}{6} \]
\[ y = \frac{4x - 24}{3} \]

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

Next, we'll rewrite both equations in slope-intercept or standard forms and compare their slopes.

Rearranging both to the form of \(y = mx + b\):

1. \(y = \frac{4}{3}x - 8\) (for first equation)
2. \(y = -\frac{4}{3}x + 8\) (for second equation)

We can see that the slopes of these two lines are different:

- First equation's slope is \(\frac{4}{3}\).
- Second equation's slope is \(-\frac{4}{3}\).

Since the slopes are different, the lines will intersect at exactly one point, meaning that the system of equations has **one unique solution**.

Therefore, the correct response is:

**one**