Here are the step-by-step solutions for each problem.
1. $x + 16 = 9x$
Step 1: Get all the $x$ terms on one side. Subtract $x$ from both sides.
$$16 = 9x - x$$
$$16 = 8x$$
Step 2: Isolate $x$ by dividing both sides by 8.
$$x = \frac{16}{8}$$
$$x = 2$$
Check:
Left Side: $2 + 16 = 18$
Right Side: $9(2) = 18$
$18 = 18$ (Correct)
2. $4y - 70 = 12y + 2$
Step 1: Move the variables to one side and numbers to the other. Subtract $4y$ from both sides.
$$-70 = 8y + 2$$
Step 2: Subtract 2 from both sides.
$$-72 = 8y$$
Step 3: Divide by 8.
$$y = \frac{-72}{8}$$
$$y = -9$$
Check:
Left Side: $4(-9) - 70 = -36 - 70 = -106$
Right Side: $12(-9) + 2 = -108 + 2 = -106$
$-106 = -106$ (Correct)
3. $5(p + 6) = 8p$
Step 1: Distribute the 5 into the parentheses.
$$5p + 30 = 8p$$
Step 2: Subtract $5p$ from both sides.
$$30 = 3p$$
Step 3: Divide by 3.
$$p = 10$$
Check:
Left Side: $5(10 + 6) = 5(16) = 80$
Right Side: $8(10) = 80$
$80 = 80$ (Correct)
4. $3(g - 7) = 2(10 + g)$
Step 1: Distribute the numbers into the parentheses.
$$3g - 21 = 20 + 2g$$
Step 2: Subtract $2g$ from both sides.
$$g - 21 = 20$$
Step 3: Add 21 to both sides.
$$g = 41$$
Check:
Left Side: $3(41 - 7) = 3(34) = 102$
Right Side: $2(10 + 41) = 2(51) = 102$
$102 = 102$ (Correct)
5. $1.8 + 7n = 9.5 - 4n$
Step 1: Add $4n$ to both sides to group the variables.
$$1.8 + 11n = 9.5$$
Step 2: Subtract 1.8 from both sides.
$$11n = 9.5 - 1.8$$
$$11n = 7.7$$
Step 3: Divide 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: $9.5 - 4(0.7) = 9.5 - 2.8 = 6.7$
$6.7 = 6.7$ (Correct)
6. $\frac{3}{7}w - 11 = -\frac{4}{7}w$
Step 1: Add $\frac{4}{7}w$ to both sides.
$$\frac{3}{7}w + \frac{4}{7}w - 11 = 0$$
$$\frac{7}{7}w - 11 = 0$$
$$1w - 11 = 0$$
$$w - 11 = 0$$
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: $-\frac{4}{7}(11) = -\frac{44}{7}$
$-\frac{44}{7} = -\frac{44}{7}$ (Correct)
7. Word Problem
Let $m$ be the number of movies.
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Club 1 Cost: $\$100$ fee + $\$10$ per movie $\rightarrow 100 + 10m$
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Club 2 Cost: $\$0$ fee + $\$15$ per movie $\rightarrow 15m$
Equation:
$$100 + 10m = 15m$$
Solve:
Subtract $10m$ from both sides:
$$100 = 5m$$
Divide by 5:
$$m = 20$$
Answer: You need to buy 20 movies for the costs 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. 20 movies
Parent Tip: Review the logic above to help your child master the concept of solving equations with variables on both sides worksheet with answer key.
