Question

Answer and show your work to these problems please :)

1. 8-3(n+4)=-5(n+4) answer should be: n=8

2. r+6(2r+6)=-6(r+2)-5r answer should be: r=-2

3. -2/3b+3/5b=3/20 answer should be: b=-9/4

4. 15/4x+15/8-3/2=6 answer should be: x=3/2

5. -11/6p+11/10=-7/5p+2/3 answer should be: p=1

6. 2/3m+11/3+7/2m=2/3m+11/3 answer should be: m=0

7. -2/3b+7/2=23/6b-37/10 answer should be: b=8/5

8. -9/4x-15/4-11/4x=3/4-1/2x answer answer should be : x=-1

9. 5/2-7/6b+5/3b=-3/2b-1/2 answer should be: x=1

10. -53/20+3/2x+5+x=x+8/5 answer should be: x=-1/2

Answer 1

Sure! Let's solve each of these problems step-by-step.

---

### 1. \( 8 - 3(n+4) = -5(n+4) \)

**Step 1:** Distribute:
\[
8 - 3n - 12 = -5n - 20
\]
**Step 2:** Simplify:
\[
-3n - 4 = -5n - 20
\]
**Step 3:** Isolate \(n\):
\[
-3n + 5n = -20 + 4
\]
\[
2n = -16
\]
\[
n = -8
\]

**Correction**: It seems there was confusion with your expected answer, but my method leads us to **\( n = -8 \)**, not \( n=8 \).

---

### 2. \( r + 6(2r + 6) = -6(r + 2) - 5r \)

**Step 1:** Distributing both sides:
\[
r + 12r + 36 = -6r - 12 - 5r
\]
\[
13r + 36 = -11r - 12
\]
**Step 2:** Combine like terms:
\[
13r + 11r = -12 - 36
\]
\[
24r = -48
\]
\[
r = -2
\]

**Answer:** \( r = -2 \)

---

### 3. \( -\frac{2}{3}b + \frac{3}{5}b = \frac{3}{20} \)

**Step 1:** Find a common denominator (which is 15):
\[
-\frac{10}{15}b + \frac{9}{15}b = \frac{3}{20}
\]
**Step 2:** Combine the left side:
\[
-\frac{b}{15} = \frac{3}{20}
\]
**Step 3:** Multiply both sides by -15:
\[
b = -\frac{3 \cdot 15}{20} = -\frac{45}{20} = -\frac{9}{4}
\]

**Answer:** \( b = -\frac{9}{4} \)

---

### 4. \( \frac{15}{4}x + \frac{15}{8} - \frac{3}{2} = 6 \)

**Step 1:** Convert \( \frac{3}{2} \) to an eighths:
\[
\frac{15}{4}x + \frac{15}{8} - \frac{12}{8} = 6
\]
**Step 2:** Simplify:
\[
\frac{15}{4}x + \frac{3}{8} = 6
\]
**Step 3:** Subtract \( \frac{3}{8} \):
\[
\frac{15}{4}x = 6 - \frac{3}{8}
\]
**Step 4:** Convert \( 6 \) to eighths:
\[
\frac{15}{4}x = \frac{48}{8} - \frac{3}{8} = \frac{45}{8}
\]
**Step 5:** Multiply by the reciprocal of \( \frac{15}{4} \):
\[
x = \frac{45}{8} \cdot \frac{4}{15} = \frac{45 \cdot 4}{8 \cdot 15} = \frac{180}{120} = \frac{3}{2}
\]

**Answer:** \( x = \frac{3}{2} \)

---

### 5. \( -\frac{11}{6}p + \frac{11}{10} = -\frac{7}{5}p + \frac{2}{3} \)

**Step 1:** Clear fractions by multiplying through by 30 (LCM of 6,10,5,3):
\[
-55p + 33 = -42p + 20
\]
**Step 2:** Combine like terms:
\[
-55p + 42p = 20 - 33
\]
\[
-13p = -13
\]
\[
p = 1
\]

**Answer:** \( p = 1 \)

---

### 6. \( \frac{2}{3}m + \frac{11}{3} + \frac{7}{2}m = \frac{2}{3}m + \frac{11}{3} \)

**Step 1:** Subtract \( \frac{2}{3}m + \frac{11}{3} \) from both sides:
\[
\frac{7}{2}m = 0
\]

**Step 2:** Clearly, this shows \( m = 0 \).

**Answer:** \( m = 0 \)

---

### 7. \( -\frac{2}{3}b + \frac{7}{2} = \frac{23}{6}b - \frac{37}{10} \)

**Step 1:** Clear the fractions:
Multiply through by 30 (LCM of denominators):
\[
-20b + 105 = 115b - 111
\]
**Step 2:** Combine:
\[
-20b - 115b = -111 - 105
\]
\[
-135b = -216
\]
\[
b = \frac{216}{135} = \frac{72}{45} = \frac{8}{5}
\]

**Answer:** \( b = \frac{8}{5} \)

---

### 8. \( -\frac{9}{4}x - \frac{15}{4} - \frac{11}{4}x = \frac{3}{4} - \frac{1}{2}x \)

**Step 1:** Combine like terms on the left:
\[
-\frac{20}{4}x - \frac{15}{4} = \frac{3}{4} - \frac{2}{4}x
\]
**Step 2:** Combine the constants and \(x\):
\[
-\frac{20}{4}x + \frac{2}{4}x = \frac{3}{4} + \frac{15}{4}
\]
**Step 3:** Isolate \(x\):
\[
-\frac{18}{4}x = \frac{18}{4}
\]
\[
x = -1
\]

**Answer:** \( x = -1 \)

---

### 9. \( \frac{5}{2} - \frac{7}{6}b + \frac{5}{3}b = -\frac{3}{2}b - \frac{1}{2} \)

**Step 1:** Combine like terms:
\[
\frac{5}{2} + \left(-\frac{7}{6} + \frac{10}{6} + \frac{9}{6}\right)b = -\frac{1}{2}
\]
Combine:
\[
\frac{5}{2} + \frac{3}{6}b = -\frac{1}{2}
\]

**Step 2:** Isolate \(b\):
\[
\frac{3}{6}b = -\frac{1}{2} - \frac{5}{2} = -3
\]
\[
b = -3 \cdot 2 = -6
\]

My calculations show instances where expected answers do not match solutions, if we work through carefully steps lead others than desired output needed.

### 10. \( -\frac{53}{20} + \frac{3}{2}x + 5 + x = x + \frac{8}{5} \)

**Step 1:** Combine like terms:
\[
-\frac{53}{20} + 5 + \frac{3}{2}x = \frac{3}{20} + 1x + \frac{8}{5}
\]

**Step 2:** Compute:
Convert \( 5 \to \frac{100}{20}, \; 8/5 = \frac{32}{20}
\]
\[
-\frac{53}{20} + \frac{100}{20} + \frac{3}{2}x = \frac{3}{20} + x + \frac{32}{20}
\]
Combine both sides:
\[
\frac{47}{20} + \frac{3}{2}x = \frac{35}{20} + x
\]
**Step 3:** And clearing \(x\):
\[
\frac{3}{2}x - x = \frac{35}{20} - \frac{47}{20}
\]
\[
\frac{3 - 2}{2}x = -\frac{12}{20}
\]
\[
\frac{1}{2}x = -0.6
\]
\[
x = -1.2 = -\frac{6}{5}
\]

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Would you like to verify one particular step further?