can you solve each equation and type the correct code? please remember to type in all caps with no spaces. you must show all work in order to earn credit! puzzle #2 1: solve for x: 5x - 3 = 3x + 1 2: solve for x: 2x + 2 = 4x - 8 3: solve for x: 10 - 5x = 14 - x 4: solve for x: 7x - 7 = -9 + 15x answer choices a: x = -4 b: x = 4 c: x = 1/2 d: x = 7 e: x = 2 f: x = -1 g: x = 1 h: x = 5 i: x = 1/4 type the 4-letter code into the answer box. all caps, no spaces.

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Accepted Answer

### Problem 1: Solve $ 5x - 3 = 3x + 1 $
# Explanation:
## Step 1: Subtract $ 3x $ from both sides
To isolate the $ x $-terms, we subtract $ 3x $ from each side. This gives us $ 5x - 3x - 3 = 3x - 3x + 1 $, which simplifies to $ 2x - 3 = 1 $.
## Step 2: Add 3 to both sides
Next, we add 3 to both sides to isolate the term with $ x $. So, $ 2x - 3 + 3 = 1 + 3 $, which simplifies to $ 2x = 4 $.
## Step 3: Divide by 2
Finally, we divide both sides by 2 to solve for $ x $. $ \frac{2x}{2} = \frac{4}{2} $, so $ x = 2 $. This corresponds to option E.

### Problem 2: Solve $ 2x + 2 = 4x - 8 $
# Explanation:
## Step 1: Subtract $ 2x $ from both sides
Subtracting $ 2x $ from each side gives $ 2x - 2x + 2 = 4x - 2x - 8 $, which simplifies to $ 2 = 2x - 8 $.
## Step 2: Add 8 to both sides
Adding 8 to both sides: $ 2 + 8 = 2x - 8 + 8 $, so $ 10 = 2x $.
## Step 3: Divide by 2
Dividing both sides by 2: $ \frac{10}{2} = \frac{2x}{2} $, so $ x = 5 $. This corresponds to option H.

### Problem 3: Solve $ 10 - 5x = 14 - x $
# Explanation:
## Step 1: Add $ 5x $ to both sides
Adding $ 5x $ to each side: $ 10 - 5x + 5x = 14 - x + 5x $, which simplifies to $ 10 = 14 + 4x $.
## Step 2: Subtract 14 from both sides
Subtracting 14 from both sides: $ 10 - 14 = 14 - 14 + 4x $, so $ -4 = 4x $.
## Step 3: Divide by 4
Dividing both sides by 4: $ \frac{-4}{4} = \frac{4x}{4} $, so $ x = -1 $. This corresponds to option F.

### Problem 4: Solve $ 7x - 7 = -9 + 15x $
# Explanation:
## Step 1: Subtract $ 7x $ from both sides
Subtracting $ 7x $ from each side gives $ 7x - 7x - 7 = -9 + 15x - 7x $, which simplifies to $ -7 = -9 + 8x $.
## Step 2: Add 9 to both sides
Adding 9 to both sides: $ -7 + 9 = -9 + 9 + 8x $, so $ 2 = 8x $.
## Step 3: Divide by 8
Dividing both sides by 8: $ \frac{2}{8} = \frac{8x}{8} $, which simplifies to $ x = \frac{1}{4} $? Wait, no, wait: $ \frac{2}{8} = \frac{1}{4} $? Wait, no, let's check again. Wait, $ 2 = 8x $, so $ x = \frac{2}{8} = \frac{1}{4} $? Wait, no, that can't be. Wait, no, let's redo step 2. Wait, original equation: $ 7x - 7 = -9 + 15x $. Subtract $ 7x $: $ -7 = -9 + 8x $. Then add 9: $ -7 + 9 = 8x $, so $ 2 = 8x $. Then $ x = \frac{2}{8} = \frac{1}{4} $? But looking at the options, I is $ x = \frac{1}{4} $. Wait, but let's check again. Wait, maybe I made a mistake. Wait, $ 7x - 7 = -9 + 15x $. Let's move $ 15x $ to the left and $ -7 $ to the right: $ 7x - 15x = -9 + 7 $, so $ -8x = -2 $, then $ x = \frac{-2}{-8} = \frac{1}{4} $. Yes, that's correct. So $ x = \frac{1}{4} $, which is option I. Wait, but let's check the options again. Wait, the options are A: -4, B:4, C:1/2, D:7, E:2, F:-1, G:1, H:5, I:1/4. So yes, $ x = \frac{1}{4} $ is I. Wait, but let's check the problem again. Wait, problem 3: $ 10 - 5x = 14 - x $. Let's solve again: $ 10 - 14 = -x + 5x $, so $ -4 = 4x $, so $ x = -1 $, which is F. Correct. Problem 4: $ 7x -7 = -9 +15x $. $ 7x -15x = -9 +7 $, $ -8x = -2 $, $ x = (-2)/(-8) = 1/4 $, which is I. So problem 4 is I.

Wait, but let's check the code. The code is the 4-letter code from the four problems. Problem 1: E, Problem 2: H, Problem 3: F, Problem 4: I? Wait, no, wait problem 4: let's check again. Wait, $ 7x -7 = -9 +15x $. Let's move terms: $ 7x -15x = -9 +7 $, $ -8x = -2 $, $ x = (-2)/(-8) = 1/4 $, which is I. So the four answers are E (problem1), H (problem2), F (problem3), I (problem4). So the code is EHFI? Wait, no, wait problem 4: let's check the options again. Wait, the options are A: x=-4, B:x=4, C:x=1/2, D:x=7, E:x=2, F:x=-1, G:x=1, H:x=5, I:x=1/4. So problem 4: x=1/4 is I. So the four answers are E (problem1), H (problem2), F (problem3), I (problem4). So the code is EHFI? Wait, but let's check problem 4 again. Wait, maybe I made a mistake. Wait, $ 7x -7 = -9 +15x $. Let's add 9 to both sides: $ 7x -7 +9 = 15x $, so $ 7x +2 = 15x $. Subtract 7x: $ 2 = 8x $, so $ x = 2/8 = 1/4 $. Correct. So problem 4 is I.

Wait, but let's check the four problems:

1. $ 5x -3 = 3x +1 $: x=2 (E)
2. $ 2x +2 = 4x -8 $: x=5 (H)
3. $ 10 -5x =14 -x $: x=-1 (F)
4. $ 7x -7 = -9 +15x $: x=1/4 (I)

So the code is EHFI. Wait, but let's check if that's correct. Wait, maybe I made a mistake in problem 4. Wait, let's solve problem 4 again:

$ 7x - 7 = -9 + 15x $

Subtract $ 7x $ from both sides:

$ -7 = -9 + 8x $

Add 9 to both sides:

$ 2 = 8x $

Divide both sides by 8:

$ x = \frac{2}{8} = \frac{1}{4} $

Yes, that's correct. So the code is EHFI.

# Answer:
EHFI

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QUESTION IMAGE

Question

Explanation:

Problem 1: Solve \( 5x - 3 = 3x + 1 \)

Step 1: Subtract \( 3x \) from both sides

Step 2: Add 3 to both sides

Step 3: Divide by 2

Problem 2: Solve \( 2x + 2 = 4x - 8 \)

Step 1: Subtract \( 2x \) from both sides

Step 2: Add 8 to both sides

Step 3: Divide by 2

Problem 3: Solve \( 10 - 5x = 14 - x \)

Step 1: Add \( 5x \) to both sides

Step 2: Subtract 14 from both sides

Step 3: Divide by 4

Problem 4: Solve \( 7x - 7 = -9 + 15x \)

Answer: