Question

How many solutions does the system of equations have? 8x−6y=48 8x+6y=48 (1 point) Responses two two none none infinite infinite one

Answer 1

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

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

We start by trying to solve the system.

**Step 1: Express both equations in a simplified form.**

From the first equation:

\[
8x - 6y = 48 \quad \Rightarrow \quad 6y = 8x - 48 \quad \Rightarrow \quad y = \frac{4}{3} x - 8
\]

From the second equation:

\[
8x + 6y = 48 \quad \Rightarrow \quad 6y = 48 - 8x \quad \Rightarrow \quad y = -\frac{4}{3} x + 8
\]

**Step 2: Analyze the equations.**

Now, we have:
1. \( y = \frac{4}{3}x - 8 \) (from the first equation)
2. \( y = -\frac{4}{3}x + 8 \) (from the second equation)

**Step 3: Find solutions.**

Next, we can see that the two equations have different slopes: the slope of the first line is \( \frac{4}{3} \) and the slope of the second line is \( -\frac{4}{3} \). Since they have different slopes, the lines are not parallel and will intersect at exactly one point.

**Conclusion:**

Thus, the system of equations has **one solution**. The correct response is **one**.