directions: solve each equation.
1. $3x + 5 = 2x - 10$
2. $4(x - 2) + 3 = 3(x + 4)$
3. $5(2y + 3) = 6y - (7 - 2y)$
4. $7 - 2z = 3z + 4(5 - z)$
5. $2(3x + 7) + x = 4x + 8$
6. $6(1 - 2m) - 3m = 2(4m + 5)$
7. $5(n + 2) - 2n = 3(2n - 1) + 7$
8. $4(2p - 3) = 2p + 6 - (p - 7)$
9. $3(q - 4) + 2q = q - 8 + 6(q + 2)$
10. $5(1 + 2r) - 3(2r - 7) = 2$
11. $7s + 2(3 - 2s) = 3(1 + s) + 14$

Question

Accepted Answer

### Problem 1: $ 3x + 5 = 2x - 10 $
# Explanation:
## Step 1: Subtract $ 2x $ from both sides
To isolate the variable terms on one side, we subtract $ 2x $ from both sides of the equation. This gives us $ 3x - 2x + 5 = 2x - 2x - 10 $, which simplifies to $ x + 5 = -10 $.
## Step 2: Subtract 5 from both sides
Next, we subtract 5 from both sides to solve for $ x $. So, $ x + 5 - 5 = -10 - 5 $, which simplifies to $ x = -15 $.
# Answer: $ x = -15 $

### Problem 2: $ 4(x - 2) + 3 = 3(x + 4) $
# Explanation:
## Step 1: Distribute the coefficients
First, we distribute the 4 on the left side and the 3 on the right side. This gives us $ 4x - 8 + 3 = 3x + 12 $.
## Step 2: Simplify both sides
Simplify the left side by combining like terms: $ 4x - 5 = 3x + 12 $.
## Step 3: Subtract $ 3x $ from both sides
Subtract $ 3x $ from both sides to get $ 4x - 3x - 5 = 3x - 3x + 12 $, which simplifies to $ x - 5 = 12 $.
## Step 4: Add 5 to both sides
Add 5 to both sides: $ x - 5 + 5 = 12 + 5 $, so $ x = 17 $.
# Answer: $ x = 17 $

### Problem 3: $ 5(2y + 3) = 6y - (7 - 2y) $
# Explanation:
## Step 1: Distribute and simplify the right side
First, distribute the 5 on the left side: $ 10y + 15 $. On the right side, simplify the expression inside the parentheses and the negative sign: $ 6y - 7 + 2y = 8y - 7 $. So the equation becomes $ 10y + 15 = 8y - 7 $.
## Step 2: Subtract $ 8y $ from both sides
Subtract $ 8y $ from both sides: $ 10y - 8y + 15 = 8y - 8y - 7 $, which simplifies to $ 2y + 15 = -7 $.
## Step 3: Subtract 15 from both sides
Subtract 15 from both sides: $ 2y + 15 - 15 = -7 - 15 $, so $ 2y = -22 $.
## Step 4: Divide by 2
Divide both sides by 2: $ \frac{2y}{2} = \frac{-22}{2} $, which gives $ y = -11 $.
# Answer: $ y = -11 $

### Problem 4: $ 7 - 2z = 3z + 4(5 - z) $
# Explanation:
## Step 1: Distribute the 4 on the right side
Distribute the 4 in $ 4(5 - z) $ to get $ 7 - 2z = 3z + 20 - 4z $.
## Step 2: Combine like terms on the right side
Combine the $ z $ terms on the right side: $ 3z - 4z = -z $, so the equation becomes $ 7 - 2z = -z + 20 $.
## Step 3: Add $ 2z $ to both sides
Add $ 2z $ to both sides: $ 7 - 2z + 2z = -z + 2z + 20 $, which simplifies to $ 7 = z + 20 $.
## Step 4: Subtract 20 from both sides
Subtract 20 from both sides: $ 7 - 20 = z + 20 - 20 $, so $ z = -13 $.
# Answer: $ z = -13 $

### Problem 5: $ 2(3x + 7) + x = 4x + 8 $
# Explanation:
## Step 1: Distribute the 2 on the left side
Distribute the 2 in $ 2(3x + 7) $ to get $ 6x + 14 + x = 4x + 8 $.
## Step 2: Combine like terms on the left side
Combine the $ x $ terms on the left side: $ 6x + x = 7x $, so the equation becomes $ 7x + 14 = 4x + 8 $.
## Step 3: Subtract $ 4x $ from both sides
Subtract $ 4x $ from both sides: $ 7x - 4x + 14 = 4x - 4x + 8 $, which simplifies to $ 3x + 14 = 8 $.
## Step 4: Subtract 14 from both sides
Subtract 14 from both sides: $ 3x + 14 - 14 = 8 - 14 $, so $ 3x = -6 $.
## Step 5: Divide by 3
Divide both sides by 3: $ \frac{3x}{3} = \frac{-6}{3} $, which gives $ x = -2 $.
# Answer: $ x = -2 $

### Problem 6: $ 6(1 - 2m) - 3m = 2(4m + 5) $
# Explanation:
## Step 1: Distribute the coefficients
Distribute the 6 on the left side and the 2 on the right side: $ 6 - 12m - 3m = 8m + 10 $.
## Step 2: Combine like terms on the left side
Combine the $ m $ terms on the left side: $ -12m - 3m = -15m $, so the equation becomes $ 6 - 15m = 8m + 10 $.
## Step 3: Add $ 15m $ to both sides
Add $ 15m $ to both sides: $ 6 - 15m + 15m = 8m + 15m + 10 $, which simplifies to $ 6 = 23m + 10 $.
## Step 4: Subtract 10 from both sides
Subtract 10 from both sides: $ 6 - 10 = 23m + 10 - 10 $, so $ -4 = 23m $.
## Step 5: Divide by 23
Divide both sides by 23: $ m = \frac{-4}{23} $.
# Answer: $ m = -\frac{4}{23} $

### Problem 7: $ 5(n + 2) - 2n = 3(2n - 1) + 7 $
# Explanation:
## Step 1: Distribute the coefficients
Distribute the 5 on the left side and the 3 on the right side: $ 5n + 10 - 2n = 6n - 3 + 7 $.
## Step 2: Combine like terms
Combine the $ n $ terms on the left side: $ 5n - 2n = 3n $, so the left side becomes $ 3n + 10 $. Combine the constant terms on the right side: $ -3 + 7 = 4 $, so the right side becomes $ 6n + 4 $. The equation is now $ 3n + 10 = 6n + 4 $.
## Step 3: Subtract $ 3n $ from both sides
Subtract $ 3n $ from both sides: $ 3n - 3n + 10 = 6n - 3n + 4 $, which simplifies to $ 10 = 3n + 4 $.
## Step 4: Subtract 4 from both sides
Subtract 4 from both sides: $ 10 - 4 = 3n + 4 - 4 $, so $ 6 = 3n $.
## Step 5: Divide by 3
Divide both sides by 3: $ \frac{6}{3} = \frac{3n}{3} $, which gives $ n = 2 $.
# Answer: $ n = 2 $

### Problem 8: $ 4(2p - 3) = 2p + 6 - (p - 7) $
# Explanation:
## Step 1: Distribute and simplify the right side
Distribute the 4 on the left side: $ 8p - 12 $. On the right side, simplify the expression with the negative sign: $ 2p + 6 - p + 7 $. Combine like terms on the right side: $ 2p - p = p $ and $ 6 + 7 = 13 $, so the right side becomes $ p + 13 $. The equation is now $ 8p - 12 = p + 13 $.
## Step 2: Subtract $ p $ from both sides
Subtract $ p $ from both sides: $ 8p - p - 12 = p - p + 13 $, which simplifies to $ 7p - 12 = 13 $.
## Step 3: Add 12 to both sides
Add 12 to both sides: $ 7p - 12 + 12 = 13 + 12 $, so $ 7p = 25 $.
## Step 4: Divide by 7
Divide both sides by 7: $ p = \frac{25}{7} $.
# Answer: $ p = \frac{25}{7} $

### Problem 9: $ 3(q - 4) + 2q = q - 8 + 6(q + 2) $
# Explanation:
## Step 1: Distribute the coefficients
Distribute the 3 on the left side and the 6 on the right side: $ 3q - 12 + 2q = q - 8 + 6q + 12 $.
## Step 2: Combine like terms
Combine the $ q $ terms on the left side: $ 3q + 2q = 5q $, so the left side becomes $ 5q - 12 $. Combine the $ q $ terms on the right side: $ q + 6q = 7q $, and combine the constant terms: $ -8 + 12 = 4 $, so the right side becomes $ 7q + 4 $. The equation is now $ 5q - 12 = 7q + 4 $.
## Step 3: Subtract $ 5q $ from both sides
Subtract $ 5q $ from both sides: $ 5q - 5q - 12 = 7q - 5q + 4 $, which simplifies to $ -12 = 2q + 4 $.
## Step 4: Subtract 4 from both sides
Subtract 4 from both sides: $ -12 - 4 = 2q + 4 - 4 $, so $ -16 = 2q $.
## Step 5: Divide by 2
Divide both sides by 2: $ \frac{-16}{2} = \frac{2q}{2} $, which gives $ q = -8 $.
# Answer: $ q = -8 $

### Problem 10: $ 5(1 + 2r) - 3(2r - 7) = 2 $
# Explanation:
## Step 1: Distribute the coefficients
Distribute the 5 and the -3: $ 5 + 10r - 6r + 21 = 2 $.
## Step 2: Combine like terms
Combine the $ r $ terms: $ 10r - 6r = 4r $, and combine the constant terms: $ 5 + 21 = 26 $. So the equation becomes $ 4r + 26 = 2 $.
## Step 3: Subtract 26 from both sides
Subtract 26 from both sides: $ 4r + 26 - 26 = 2 - 26 $, which simplifies to $ 4r = -24 $.
## Step 4: Divide by 4
Divide both sides by 4: $ \frac{4r}{4} = \frac{-24}{4} $, so $ r = -6 $.
# Answer: $ r = -6 $

### Problem 11: $ 7s + 2(3 - 2s) = 3(1 + s) + 14 $
# Explanation:
## Step 1: Distribute the coefficients
Distribute the 2 on the left side and the 3 on the right side: $ 7s + 6 - 4s = 3 + 3s + 14 $.
## Step 2: Combine like terms
Combine the $ s $ terms on the left side: $ 7s - 4s = 3s $, so the left side becomes $ 3s + 6 $. Combine the constant terms on the right side: $ 3 + 14 = 17 $, so the right side becomes $ 3s + 17 $. The equation is now $ 3s + 6 = 3s + 17 $.
## Step 3: Subtract $ 3s $ from both sides
Subtract $ 3s $ from both sides: $ 3s - 3s + 6 = 3s - 3s + 17 $, which simplifies to $ 6 = 17 $.
Since $ 6
eq 17 $, this equation has no solution.
# Answer: No solution

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( 3x + 5 = 2x - 10 \)

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

Step 2: Subtract 5 from both sides

Step 1: Distribute the coefficients

Step 2: Simplify both sides

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

Step 4: Add 5 to both sides

Step 1: Distribute and simplify the right side

Step 2: Subtract \( 8y \) from both sides

Step 3: Subtract 15 from both sides

Step 4: Divide by 2

Answer:

Problem 2: \( 4(x - 2) + 3 = 3(x + 4) \)