Questions: 1. 7c + 12 = -4c + 78 2. 2m - 13 = -8m + 27 3. 9x - 4 = 2x + 3 4. 6 + 3t = 8t - 14 5. (b - 4) / 6 = b / 2 6. (3v + 12) / 6 = 4v / 3 7. 2(r + 6) = 4(r + 4) 8. 6(n + 5) = 3(n + 16) 9. 5(g + 8) - 7 = 117 - g 10. 12 - 4 / 5(x + 15) = 2 / 5 x + 6 11. 3(3m - 2) = 2(3m + 3) 12. 6(3a + 1) - 30 = 3(2a - 4) 13. 7n + 6 = 4n - 9 14. -6(2r + 8) = -10(r - 3) 15. 5 - 3(w + 4) = w - 7 16. 2x - 5(x - 3) = 2(x - 10) 17. In the 2010 Winter Olympic Games, the United States won 1 more than 4 times the number of gold medals France won. The United States won 7 more gold metals than France. Write and solve an equation to find the number of gold medals each country won. 18. Diego's sister is twice his age minus 9 years. She is also as old as half the sum of the ages of Diego and both of his 12 -year-old twin brothers. Write and solve an equation to find the ages of Diego and his sister. 19. The table shows the current heights and average growth rates of two different species of trees. Write and solve an equation to find how long it will take for the two trees to be the same height. Tree Species Current Height Annual growth A 38 inches 4 inches B 45.5 inches 2.5 inches 20. A dog weighs two pounds less than three times the weight of a cat. The dog also weighs twenty-two pounds more than the cat. Write and solve an equation to find the weights of the dog and the cat.

1. 7c + 12 = -4c + 78
2. 2m - 13 = -8m + 27
3. 9x - 4 = 2x + 3
4. 6 + 3t = 8t - 14
5. (b - 4) / 6 = b / 2
6. (3v + 12) / 6 = 4v / 3
7. 2(r + 6) = 4(r + 4)
8. 6(n + 5) = 3(n + 16)
9. 5(g + 8) - 7 = 117 - g
10. 12 - 4 / 5(x + 15) = 2 / 5 x + 6
11. 3(3m - 2) = 2(3m + 3)
12. 6(3a + 1) - 30 = 3(2a - 4)
13. 7n + 6 = 4n - 9
14. -6(2r + 8) = -10(r - 3)
15. 5 - 3(w + 4) = w - 7
16. 2x - 5(x - 3) = 2(x - 10)
17. In the 2010 Winter Olympic Games, the United States won 1 more than 4 times the number of gold medals France won. The United States won 7 more gold metals than France. Write and solve an equation to find the number of gold medals each country won.
18. Diego's sister is twice his age minus 9 years. She is also as old as half the sum of the ages of Diego and both of his 12 -year-old twin brothers. Write and solve an equation to find the ages of Diego and his sister.
19. The table shows the current heights and average growth rates of two different species of trees. Write and solve an equation to find how long it will take for the two trees to be the same height.

Tree Species  Current Height  Annual growth
A  38 inches  4 inches
B  45.5 inches  2.5 inches

20. A dog weighs two pounds less than three times the weight of a cat. The dog also weighs twenty-two pounds more than the cat. Write and solve an equation to find the weights of the dog and the cat.
Transcript text: 1. $7 c+12=-4 c+78$ 2. $2 m-13=-8 m+27$ 3. $9 x-4=2 x+3$ 4. $6+3 t=8 t-14$ 5. $\frac{b-4}{6}=\frac{b}{2}$ 6. $\frac{3 v+12}{6}=\frac{4 v}{3}$ 7. $2(r+6)=4(r+4)$ 8. $6(n+5)=3(n+16)$ 9. $5(g+8)-7=117-g$ 10. $12-\frac{4}{5}(x+15)=\frac{2}{5} x+6$ 11. $3(3 m-2)=2(3 m+3)$ 12. $6(3 a+1)-30=3(2 a-4)$ 13. $7 n+6=4 n-9$ 14. $-6(2 r+8)=-10(r-3)$ 15. $5-3(w+4)=w-7$ 16. $2 x-5(x-3)=2(x-10)$ 17. OLYMPICS In the 2010 Winter Olympic Games, the United States won 1 more than 4 times the number of gold medals France won. The United States won 7 more gold metals than France. Write and solve an equation to find the number of gold medals each country won. 18. REASONING Diego's sister is twice his age minus 9 years. She is also as old as half the sum of the ages of Diego and both of his 12 -year-old twin brothers. Write and solve an equation to find the ages of Diego and his sister. 19. NATURE The table shows the current heights and average growth rates of two different species of trees. Write and solve an equation to find how long it will take for the two trees to be the same height. \begin{tabular}{|l|l|l|} \hline Tree Species & Current Height & Annual growth \\ \hline A & 38 inches & 4 inches \\ \hline B & 45.5 inches & 2.5 inches \\ \hline \end{tabular} 20. WEIGHT A dog weighs two pounds less than three times the weight of a cat. The dog also weighs twenty-two pounds more than the cat. Write and solve an equation to find the weights of the dog and the cat.
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Solution

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Solution Steps

Solution Approach
  1. For the equation \(7c + 12 = -4c + 78\), we will first move all terms involving \(c\) to one side and constant terms to the other side. Then, solve for \(c\).

  2. For the equation \(2m - 13 = -8m + 27\), we will move all terms involving \(m\) to one side and constant terms to the other side. Then, solve for \(m\).

  3. For the equation \(9x - 4 = 2x + 3\), we will move all terms involving \(x\) to one side and constant terms to the other side. Then, solve for \(x\).

Step 1: Solve for \( c \)

We start with the equation: \[ 7c + 12 = -4c + 78 \] Rearranging the equation gives: \[ 7c + 4c = 78 - 12 \] This simplifies to: \[ 11c = 66 \] Dividing both sides by 11, we find: \[ c = 6 \]

Step 2: Solve for \( m \)

Next, we solve the equation: \[ 2m - 13 = -8m + 27 \] Rearranging the equation gives: \[ 2m + 8m = 27 + 13 \] This simplifies to: \[ 10m = 40 \] Dividing both sides by 10, we find: \[ m = 4 \]

Step 3: Solve for \( x \)

Finally, we solve the equation: \[ 9x - 4 = 2x + 3 \] Rearranging the equation gives: \[ 9x - 2x = 3 + 4 \] This simplifies to: \[ 7x = 7 \] Dividing both sides by 7, we find: \[ x = 1 \]

Final Answer

The solutions are: \[ \boxed{c = 6}, \quad \boxed{m = 4}, \quad \boxed{x = 1} \]

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