problems 1–3: solve each equation. show your thinking.
1. $2(x + 5) = 3x + 1$
2. $3y - 4 = 6 - 2y$
3. $3(n + 2) = 9(6 - n)$

4. here is how gabriela solved an equation. circle the mistakes and explain them.
gabriela
$12(5 + 2y) = 4y - (5 - 9y)$
$72 + 24y = 4y - 5 - 9y$
$72 + 24y = -5y - 5$
$24y = -5y - 77$
$29y = -77$
$y = -\frac{77}{29}$
what is the correct solution?

problems 5–6: solve each equation. show your thinking.
5. $\frac{1}{9}(2m - 16) = \frac{1}{3}(2m + 4)$
6. $1.5(5 + 0.2y) = 0.4y - (0.6 - 0.9y)$

Question

problems 1–3: solve each equation. show your thinking.
1. $2(x + 5) = 3x + 1$
2. $3y - 4 = 6 - 2y$
3. $3(n + 2) = 9(6 - n)$

4. here is how gabriela solved an equation. circle the mistakes and explain them.
gabriela
$12(5 + 2y) = 4y - (5 - 9y)$
$72 + 24y = 4y - 5 - 9y$
$72 + 24y = -5y - 5$
$24y = -5y - 77$
$29y = -77$
$y = -\frac{77}{29}$
what is the correct solution?

problems 5–6: solve each equation. show your thinking.
5. $\frac{1}{9}(2m - 16) = \frac{1}{3}(2m + 4)$
6. $1.5(5 + 0.2y) = 0.4y - (0.6 - 0.9y)$

Accepted Answer

### Problem 1: Solve $ 2(x + 5) = 3x + 1 $
# Explanation:
## Step 1: Distribute the left side
$ 2x + 10 = 3x + 1 $
## Step 2: Subtract $ 2x $ from both sides
$ 10 = x + 1 $
## Step 3: Subtract 1 from both sides
$ x = 9 $
# Answer:
$ x = 9 $

### Problem 2: Solve $ 3y - 4 = 6 - 2y $
# Explanation:
## Step 1: Add $ 2y $ to both sides
$ 5y - 4 = 6 $
## Step 2: Add 4 to both sides
$ 5y = 10 $
## Step 3: Divide by 5
$ y = 2 $
# Answer:
$ y = 2 $

### Problem 3: Solve $ 3(n + 2) = 9(6 - n) $
# Explanation:
## Step 1: Distribute both sides
$ 3n + 6 = 54 - 9n $
## Step 2: Add $ 9n $ to both sides
$ 12n + 6 = 54 $
## Step 3: Subtract 6 from both sides
$ 12n = 48 $
## Step 4: Divide by 12
$ n = 4 $
# Answer:
$ n = 4 $

### Problem 4: Analyze Gabriela's Solution and Find Correct Solution
#### Mistakes in Gabriela's Work:
1. **First Step (Distributive Property):**
   $ 12(5 + 2y) = 4y - (5 - 9y) $
   Correct distribution: $ 12 \times 5 + 12 \times 2y = 60 + 24y $, but Gabriela wrote $ 72 + 24y $ (incorrect multiplication: $ 12 \times 5 = 60 $, not 72).
   Also, when distributing the negative sign on the right: $ 4y - 5 + 9y $ (because $ -(5 - 9y) = -5 + 9y $), but Gabriela wrote $ 4y - 5 - 9y $ (incorrect sign on $ -9y $).

2. **Third Step (Subtracting 72):**
   From $ 72 + 24y = -5y - 5 $, subtracting 72 gives $ 24y = -5y - 5 - 72 $, so $ 24y = -5y - 77 $ (Gabriela’s step here is correct in operation, but the initial 72 was wrong).

3. **Fourth Step (Adding $ 5y $):**
   From $ 24y = -5y - 77 $, adding $ 5y $ gives $ 29y = -77 $ (correct operation, but the earlier mistake propagates).

#### Correct Solution:
Start with $ 12(5 + 2y) = 4y - (5 - 9y) $
## Step 1: Distribute both sides
$ 60 + 24y = 4y - 5 + 9y $
## Step 2: Combine like terms on the right
$ 60 + 24y = 13y - 5 $
## Step 3: Subtract $ 13y $ from both sides
$ 60 + 11y = -5 $
## Step 4: Subtract 60 from both sides
$ 11y = -65 $
## Step 5: Divide by 11
$ y = -\frac{65}{11} $ (or $ y \approx -5.91 $)

### Problem 5: Solve $ \frac{1}{9}(2m - 16) = \frac{1}{3}(2m + 4) $
# Explanation:
## Step 1: Multiply both sides by 9 to eliminate denominators
$ 2m - 16 = 3(2m + 4) $
## Step 2: Distribute the right side
$ 2m - 16 = 6m + 12 $
## Step 3: Subtract $ 2m $ from both sides
$ -16 = 4m + 12 $
## Step 4: Subtract 12 from both sides
$ -28 = 4m $
## Step 5: Divide by 4
$ m = -7 $
# Answer:
$ m = -7 $

### Problem 6: Solve $ 1.5(5 + 0.2y) = 0.4y - (0.6 - 0.9y) $
# Explanation:
## Step 1: Distribute both sides
$ 7.5 + 0.3y = 0.4y - 0.6 + 0.9y $
## Step 2: Combine like terms on the right
$ 7.5 + 0.3y = 1.3y - 0.6 $
## Step 3: Subtract $ 0.3y $ from both sides
$ 7.5 = 1.0y - 0.6 $
## Step 4: Add 0.6 to both sides
$ 8.1 = y $
# Answer:
$ y = 8.1 $

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QUESTION IMAGE

Question

Explanation:

Problem 1: Solve \( 2(x + 5) = 3x + 1 \)

Step 1: Distribute the left side

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

Step 3: Subtract 1 from both sides

Step 1: Add \( 2y \) to both sides

Step 2: Add 4 to both sides

Step 3: Divide by 5

Step 1: Distribute both sides

Step 2: Add \( 9n \) to both sides

Step 3: Subtract 6 from both sides

Step 4: Divide by 12

Answer:

Problem 2: Solve \( 3y - 4 = 6 - 2y \)