Comprehensive algebra practice sheet featuring multi-step equations and a cost-comparison word problem.

Here are the step-by-step solutions for each problem.

1. $x + 16 = 9x$

Step 1: Subtract $x$ from both sides to group the $x$ terms together.
$$16 = 9x - x$$
$$16 = 8x$$

Step 2: Divide both sides by 8 to solve for $x$.
$$\frac{16}{8} = \frac{8x}{8}$$
$$2 = x$$

Check:
Left Side: $2 + 16 = 18$
Right Side: $9(2) = 18$
Since $18 = 18$, the solution is correct.

2. $4y - 70 = 12y + 2$

Step 1: Subtract $4y$ from both sides to move variables to one side.
$$-70 = 12y - 4y + 2$$
$$-70 = 8y + 2$$

Step 2: Subtract 2 from both sides to isolate the term with $y$.
$$-70 - 2 = 8y$$
$$-72 = 8y$$

Step 3: Divide both sides by 8.
$$\frac{-72}{8} = y$$
$$-9 = y$$

Check:
Left Side: $4(-9) - 70 = -36 - 70 = -106$
Right Side: $12(-9) + 2 = -108 + 2 = -106$
Since $-106 = -106$, the solution is correct.

3. $5(p + 6) = 8p$

Step 1: Distribute the 5 into the parentheses.
$$5p + 30 = 8p$$

Step 2: Subtract $5p$ from both sides.
$$30 = 8p - 5p$$
$$30 = 3p$$

Step 3: Divide both sides by 3.
$$\frac{30}{3} = p$$
$$10 = p$$

Check:
Left Side: $5(10 + 6) = 5(16) = 80$
Right Side: $8(10) = 80$
Since $80 = 80$, the solution is correct.

4. $3(g - 7) = 2(10 + g)$

Step 1: Distribute the numbers outside the parentheses.
$$3g - 21 = 20 + 2g$$

Step 2: Subtract $2g$ from both sides.
$$3g - 2g - 21 = 20$$
$$g - 21 = 20$$

Step 3: Add 21 to both sides.
$$g = 20 + 21$$
$$g = 41$$

Check:
Left Side: $3(41 - 7) = 3(34) = 102$
Right Side: $2(10 + 41) = 2(51) = 102$
Since $102 = 102$, the solution is correct.

5. $1.8 + 7n = 9.5 - 4n$

Step 1: Add $4n$ to both sides to group the variable terms.
$$1.8 + 7n + 4n = 9.5$$
$$1.8 + 11n = 9.5$$

Step 2: Subtract 1.8 from both sides.
$$11n = 9.5 - 1.8$$
$$11n = 7.7$$

Step 3: Divide both sides by 11.
$$n = \frac{7.7}{11}$$
$$n = 0.7$$

Check:
Left Side: $1.8 + 7(0.7) = 1.8 + 4.9 = 6.7$
Right Side: $9.5 - 4(0.7) = 9.5 - 2.8 = 6.7$
Since $6.7 = 6.7$, the solution is correct.

6. $\frac{3}{7}w - 11 = -\frac{4}{7}w$

Step 1: Add $\frac{4}{7}w$ to both sides to group the $w$ terms.
$$\frac{3}{7}w + \frac{4}{7}w - 11 = 0$$
$$\frac{7}{7}w - 11 = 0$$
$$1w - 11 = 0$$
$$w - 11 = 0$$

Step 2: Add 11 to both sides.
$$w = 11$$

Check:
Left Side: $\frac{3}{7}(11) - 11 = \frac{33}{7} - \frac{77}{7} = -\frac{44}{7}$
Right Side: $-\frac{4}{7}(11) = -\frac{44}{7}$
Since $-\frac{44}{7} = -\frac{44}{7}$, the solution is correct.

7. Word Problem

Define the variable:
Let $m$ be the number of movies.

Write expressions for the cost of each club:
* Club 1: $\$100$ fee + $\$10$ per movie $\rightarrow 100 + 10m$
* Club 2: $\$0$ fee + $\$15$ per movie $\rightarrow 15m$

Set up the equation:
To find when the costs are the same, set the expressions equal to each other:
$$100 + 10m = 15m$$

Solve the equation:
1. Subtract $10m$ from both sides:
$$100 = 15m - 10m$$
$$100 = 5m$$

2. Divide by 5:
$$m = \frac{100}{5}$$
$$m = 20$$

Answer: You need to buy 20 movies for the cost to be the same.

Final Answer:
1. $x = 2$
2. $y = -9$
3. $p = 10$
4. $g = 41$
5. $n = 0.7$
6. $w = 11$
7. Equation: $100 + 10m = 15m$; Solution: 20 movies