Question

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

none
none

two
two

infinite
infinite

one

Answer 1

are you sure

Answer 2

To determine how many solutions the system of equations has, let's first analyze the given equations:

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

We can simplify and analyze these equations by manipulating them.

### Step 1: Solve for \( y \) in terms of \( x \).

**Equation 1**:
\[
8x - 6y = 48 \implies 6y = 8x - 48 \implies y = \frac{8x - 48}{6} \implies y = \frac{4x - 24}{3}
\]

**Equation 2**:
\[
8x + 6y = 48 \implies 6y = 48 - 8x \implies y = \frac{48 - 8x}{6} \implies y = 8 - \frac{4x}{3}
\]

### Step 2: Set the equations for \( y \) equal to each other:

\[
\frac{4x - 24}{3} = 8 - \frac{4x}{3}
\]

### Step 3: Clear the fractions by multiplying everything by 3:

\[
4x - 24 = 24 - 4x
\]

### Step 4: Combine like terms:

\[
4x + 4x = 24 + 24
\]

\[
8x = 48 \implies x = 6
\]

### Step 5: Substitute \( x \) back into one of the equations to find \( y \):

Using **Equation 1**:
\[
8(6) - 6y = 48 \implies 48 - 6y = 48 \implies -6y = 0 \implies y = 0
\]

### Conclusion:
The system has one solution: \( (x, y) = (6, 0) \).

### Final Answer:
**one**

Answer 3

Let's double-check the analysis step by step to ensure we have the correct answer.

The system of equations is:

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

### Step 1: Look at the equations closely.

We can add both equations together to see if they reveal something:

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

This simplifies to:

\[
16x = 96 \implies x = 6
\]

### Step 2: Substitute \( x = 6 \) back into either equation to find \( y \).

Using **Equation 1**:

\[
8(6) - 6y = 48
\]
\[
48 - 6y = 48 \implies -6y = 0 \implies y = 0
\]

So we have the solution \( (x, y) = (6, 0) \).

### Step 3: Check for consistency of the equations.

Now, let's look deeper into the nature of the given equations:

- The first equation can be rewritten as \( y = \frac{4}{3}x - 8 \).
- The second equation can be rewritten as \( y = -\frac{4}{3}x + 8 \).

These two equations represent different lines because they have different slopes (\(\frac{4}{3}\) for the first and \(-\frac{4}{3}\) for the second). Since the lines intersect at a single point \( (6, 0) \), the system has exactly one solution.

### Conclusion:
After careful verification, the system of equations has **one solution**. The initial analysis is confirmed to be accurate.

The final answer is:
**one**