for each linear equation in the table, indicate whether the equation has no solution, one solution, or infinitely many solutions. use the drop - down menus to choose the correct response for each equation. equation no solution/ one solution/ infinitely many solutions $5x + 13 = 13$ no solution $x - 3 = x + 3$ one solution $-4x - 8 = 4x + 8$ one solution

Question

Accepted Answer

To determine the number of solutions for each linear equation, we analyze them one by one:

### Equation 1: $ 5x + 13 = 13 $
## Step 1: Subtract 13 from both sides
Subtract 13 from both sides of the equation:
$ 5x + 13 - 13 = 13 - 13 $
Simplify: $ 5x = 0 $

## Step 2: Solve for $ x $
Divide both sides by 5:
$ \frac{5x}{5} = \frac{0}{5} $
Simplify: $ x = 0 $

Since we found a unique value for $ x $, this equation has **one solution**.

### Equation 2: $ x - 3 = x + 3 $
## Step 1: Subtract $ x $ from both sides
Subtract $ x $ from both sides:
$ x - 3 - x = x + 3 - x $
Simplify: $ -3 = 3 $

The statement $ -3 = 3 $ is false. There are no values of $ x $ that satisfy this equation, so it has **no solution**.

### Equation 3: $ -4x - 8 = 4x + 8 $
## Step 1: Add $ 4x $ to both sides
Add $ 4x $ to both sides:
$ -4x - 8 + 4x = 4x + 8 + 4x $
Simplify: $ -8 = 8x + 8 $

## Step 2: Subtract 8 from both sides
Subtract 8 from both sides:
$ -8 - 8 = 8x + 8 - 8 $
Simplify: $ -16 = 8x $

## Step 3: Solve for $ x $
Divide both sides by 8:
$ \frac{-16}{8} = \frac{8x}{8} $
Simplify: $ -2 = x $

Since we found a unique value for $ x $, this equation has **one solution**? Wait, no—wait, let’s recheck:

Wait, $ -4x - 8 = 4x + 8 $
Add $ 4x $ to both sides: $ -8 = 8x + 8 $
Subtract 8: $ -16 = 8x $
Divide by 8: $ x = -2 $. So it does have one solution. Wait, but let’s confirm:

Wait, no—wait, the original equation: $ -4x - 8 = 4x + 8 $
If $ x = -2 $, substitute back:
Left side: $ -4(-2) - 8 = 8 - 8 = 0 $
Right side: $ 4(-2) + 8 = -8 + 8 = 0 $
So $ x = -2 $ works. So it has one solution. Wait, but maybe I made a mistake earlier? Wait, no—let’s re-express:

Wait, the initial dropdown for $ -4x - 8 = 4x + 8 $ was set to “one solution,” which is correct. Wait, but let’s re-express the steps:

Wait, $ -4x - 8 = 4x + 8 $
Bring all $ x $ terms to one side: $ -4x - 4x = 8 + 8 $
$ -8x = 16 $
$ x = -2 $. So yes, one solution.

### Final Answers:
- $ 5x + 13 = 13 $: One Solution
- $ x - 3 = x + 3 $: No Solution
- $ -4x - 8 = 4x + 8 $: One Solution

To summarize:
1. $ 5x + 13 = 13 $: One Solution
2. $ x - 3 = x + 3 $: No Solution
3. $ -4x - 8 = 4x + 8 $: One Solution

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QUESTION IMAGE

Question

Explanation:

Equation 1: \( 5x + 13 = 13 \)

Step 1: Subtract 13 from both sides

Step 2: Solve for \( x \)

Equation 2: \( x - 3 = x + 3 \)

Step 1: Subtract \( x \) from both sides

Equation 3: \( -4x - 8 = 4x + 8 \)

Step 1: Add \( 4x \) to both sides

Step 2: Subtract 8 from both sides

Step 3: Solve for \( x \)

Final Answers:

Answer: