algebra 1
week 6 solving practice
solve each equation.
1) $8 - 6v=5v - 3$
2) $5n - 6 = 6n+1$
3) $5x=x - 6x$
4) $2k + 7k=4 + 4k-3 - 1$
5) $a + 2=14 + 3a$
6) $-3x=x + 8$
7) $6b - 6=8(b + 1)$
8) $39 + 2a=7 + 8(4 + 8a)$
9) $-5 - p=-2(p + 3)$
10) $-39 + 6x=7(6 + 4x)+5x$
11) $4 + 5a=4(1 - 8a)$
12) $-5p + 3=-2(p + 6)$
13) $-v + v=4$
14) $4n - 7 - 4n=0$
15) $p - 8 - p=-6$
16) $-5p + 5p=-2$
17) $0=-2v+2v$
18) $8 - 7r + 7r=13$

Question

Accepted Answer

1. **For the equation $8 - 6v=5v - 3$**:
   - **Step 1: Add $6v$ to both sides**
     - The goal is to get all the variable - terms on one side. $8-6v + 6v=5v-3 + 6v$, which simplifies to $8 = 11v-3$.
   - **Step 2: Add 3 to both sides**
     - To isolate the term with the variable, $8 + 3=11v-3 + 3$, resulting in $11 = 11v$.
   - **Step 3: Divide both sides by 11**
     - $\frac{11}{11}=\frac{11v}{11}$, so $v = 1$.
2. **For the equation $5n-6 = 6n + 1$**:
   - **Step 1: Subtract $5n$ from both sides**
     - $5n-6-5n=6n + 1-5n$, giving $-6=n + 1$.
   - **Step 2: Subtract 1 from both sides**
     - $-6-1=n + 1-1$, so $n=-7$.
3. **For the equation $5x=x - 6x$**:
   - **Step 1: Simplify the right - hand side**
     - Combine like terms on the right - hand side: $x-6x=-5x$, so the equation becomes $5x=-5x$.
   - **Step 2: Add $5x$ to both sides**
     - $5x + 5x=-5x+5x$, which gives $10x = 0$.
   - **Step 3: Divide both sides by 10**
     - $\frac{10x}{10}=\frac{0}{10}$, so $x = 0$.
4. **For the equation $2k + 7k=4 + 4k-3-1$**:
   - **Step 1: Simplify both sides**
     - Combine like terms on the left - hand side: $2k + 7k = 9k$. Combine like terms on the right - hand side: $4 + 4k-3-1=4k+(4 - 3-1)=4k$. So the equation is $9k=4k$.
   - **Step 2: Subtract $4k$ from both sides**
     - $9k-4k=4k-4k$, resulting in $5k = 0$.
   - **Step 3: Divide both sides by 5**
     - $\frac{5k}{5}=\frac{0}{5}$, so $k = 0$.
5. **For the equation $a + 2=14+3a$**:
   - **Step 1: Subtract $a$ from both sides**
     - $a + 2-a=14+3a-a$, giving $2=14 + 2a$.
   - **Step 2: Subtract 14 from both sides**
     - $2-14=14 + 2a-14$, so $-12 = 2a$.
   - **Step 3: Divide both sides by 2**
     - $\frac{-12}{2}=\frac{2a}{2}$, so $a=-6$.
6. **For the equation $-3x=x + 8$**:
   - **Step 1: Subtract $x$ from both sides**
     - $-3x-x=x + 8-x$, resulting in $-4x=8$.
   - **Step 2: Divide both sides by $-4$**
     - $\frac{-4x}{-4}=\frac{8}{-4}$, so $x=-2$.
7. **For the equation $6b-6 = 8(b + 1)$**:
   - **Step 1: Expand the right - hand side**
     - Using the distributive property $8(b + 1)=8b+8$, so the equation is $6b-6=8b + 8$.
   - **Step 2: Subtract $6b$ from both sides**
     - $6b-6-6b=8b + 8-6b$, giving $-6=2b + 8$.
   - **Step 3: Subtract 8 from both sides**
     - $-6-8=2b + 8-8$, so $-14 = 2b$.
   - **Step 4: Divide both sides by 2**
     - $\frac{-14}{2}=\frac{2b}{2}$, so $b=-7$.
8. **For the equation $39 + 2a=7+8(4 + 8a)$**:
   - **Step 1: Expand the right - hand side**
     - Using the distributive property $8(4 + 8a)=32+64a$, so the equation is $39 + 2a=7+32+64a$.
   - **Step 2: Simplify the right - hand side**
     - Combine like terms on the right - hand side: $7+32 = 39$, so $39 + 2a=39+64a$.
   - **Step 3: Subtract $2a$ from both sides**
     - $39 + 2a-2a=39+64a-2a$, giving $39=39 + 62a$.
   - **Step 4: Subtract 39 from both sides**
     - $39-39=39 + 62a-39$, so $0 = 62a$.
   - **Step 5: Divide both sides by 62**
     - $\frac{0}{62}=\frac{62a}{62}$, so $a = 0$.
9. **For the equation $-5-p=-2(p + 3)$**:
   - **Step 1: Expand the right - hand side**
     - Using the distributive property $-2(p + 3)=-2p-6$, so the equation is $-5-p=-2p-6$.
   - **Step 2: Add $2p$ to both sides**
     - $-5-p+2p=-2p-6+2p$, giving $-5 + p=-6$.
   - **Step 3: Add 5 to both sides**
     - $-5 + p+5=-6+5$, so $p=-1$.
10. **For the equation $-39 + 6x=7(6 + 4x)+5x$**:
   - **Step 1: Expand the right - hand side**
     - Using the distributive property $7(6 + 4x)=42+28x$, so the equation is $-39 + 6x=42+28x+5x$.
   - **Step 2: Combine like terms on the right - hand side**
     - $28x+5x = 33x$, so $-39 + 6x=42+33x$.
   - **Step 3: Subtract $6x$ from both sides**
     - $-39 + 6x-6x=42+33x-6x$, giving $-39=42 + 27x$.
   - **Step 4: Subtract 42 from both sides**
     - $-39-42=42 + 27x-42$, so $-81 = 27x$.
   - **Step 5: Divide both sides by 27**
     - $\frac{-81}{27}=\frac{27x}{27}$, so $x=-3$.
11. **For the equation $4 + 5a=4(1-8a)$**:
   - **Step 1: Expand the right - hand side**
     - Using the distributive property $4(1-8a)=4-32a$, so the equation is $4 + 5a=4-32a$.
   - **Step 2: Subtract 4 from both sides**
     - $4 + 5a-4=4-32a-4$, giving $5a=-32a$.
   - **Step 3: Add $32a$ to both sides**
     - $5a+32a=-32a+32a$, resulting in $37a = 0$.
   - **Step 4: Divide both sides by 37**
     - $\frac{37a}{37}=\frac{0}{37}$, so $a = 0$.
12. **For the equation $-5p + 3=-2(p + 6)$**:
   - **Step 1: Expand the right - hand side**
     - Using the distributive property $-2(p + 6)=-2p-12$, so the equation is $-5p + 3=-2p-12$.
   - **Step 2: Add $5p$ to both sides**
     - $-5p + 3+5p=-2p-12+5p$, giving $3 = 3p-12$.
   - **Step 3: Add 12 to both sides**
     - $3+12=3p-12+12$, so $15 = 3p$.
   - **Step 4: Divide both sides by 3**
     - $\frac{15}{3}=\frac{3p}{3}$, so $p = 5$.
13. **For the equation $-v + v=4$**:
   - **Step 1: Simplify the left - hand side**
     - Combine like terms: $-v + v = 0$, and $0
eq4$, so there is no solution.
14. **For the equation $4n-7-4n=0$**:
   - **Step 1: Simplify the left - hand side**
     - Combine like terms: $4n-4n-7=-7$, and $-7
eq0$, so there is no solution.
15. **For the equation $p-8-p=-6$**:
   - **Step 1: Simplify the left - hand side**
     - Combine like terms: $p-8-p=-8$, and $-8
eq - 6$, so there is no solution.
16. **For the equation $-5p + 5p=-2$**:
   - **Step 1: Simplify the left - hand side**
     - Combine like terms: $-5p + 5p = 0$, and $0
eq-2$, so there is no solution.
17. **For the equation $0=-2v+2v$**:
   - **Step 1: Simplify the right - hand side**
     - Combine like terms: $-2v + 2v = 0$, so the equation $0 = 0$ is an identity, and $v$ can be any real number.
18. **For the equation $8-7r + 7r=13$**:
   - **Step 1: Simplify the left - hand side**
     - Combine like terms: $-7r + 7r=0$, so the left - hand side is 8, and $8
eq13$, so there is no solution.

# Answer:
1. $v = 1$
2. $n=-7$
3. $x = 0$
4. $k = 0$
5. $a=-6$
6. $x=-2$
7. $b=-7$
8. $a = 0$
9. $p=-1$
10. $x=-3$
11. $a = 0$
12. $p = 5$
13. No solution
14. No solution
15. No solution
16. No solution
17. $v$ is any real number
18. No solution

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