Algebra worksheet for solving equations to find the value of y.

Problem Overview:

The worksheet contains 20 algebraic equations where the goal is to solve for the variable \( y \). Each equation involves either multiplication, addition, subtraction, or a combination of these operations. We will solve each equation step by step.

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Solutions:

#### Part 1: Simple Multiplication Equations
These equations are of the form \( ay = b \), where we solve for \( y \) by dividing both sides by \( a \).

1. \( 3y = 27 \)
\[
y = \frac{27}{3} = 9
\]
Answer: \( y = 9 \)

2. \( 9y = 81 \)
\[
y = \frac{81}{9} = 9
\]
Answer: \( y = 9 \)

3. \( 12y = 132 \)
\[
y = \frac{132}{12} = 11
\]
Answer: \( y = 11 \)

4. \( 6y = 72 \)
\[
y = \frac{72}{6} = 12
\]
Answer: \( y = 12 \)

5. \( 20y = 120 \)
\[
y = \frac{120}{20} = 6
\]
Answer: \( y = 6 \)

6. \( 15y = 75 \)
\[
y = \frac{75}{15} = 5
\]
Answer: \( y = 5 \)

7. \( 10y = 160 \)
\[
y = \frac{160}{10} = 16
\]
Answer: \( y = 16 \)

8. \( 9y = 72 \)
\[
y = \frac{72}{9} = 8
\]
Answer: \( y = 8 \)

9. \( 11y = 88 \)
\[
y = \frac{88}{11} = 8
\]
Answer: \( y = 8 \)

10. \( 4y = 36 \)
\[
y = \frac{36}{4} = 9
\]
Answer: \( y = 9 \)

---

#### Part 2: Equations Involving Addition and Subtraction
These equations are of the form \( ay + b = c \) or \( ay - b = c \). We solve for \( y \) by isolating it on one side of the equation.

11. \( 5y + 15 = 65 \)
\[
5y = 65 - 15 \quad \text{(Subtract 15 from both sides)}
\]
\[
5y = 50
\]
\[
y = \frac{50}{5} = 10
\]
Answer: \( y = 10 \)

12. \( 9y + 20 = 83 \)
\[
9y = 83 - 20 \quad \text{(Subtract 20 from both sides)}
\]
\[
9y = 63
\]
\[
y = \frac{63}{9} = 7
\]
Answer: \( y = 7 \)

13. \( 7y + 9 = 44 \)
\[
7y = 44 - 9 \quad \text{(Subtract 9 from both sides)}
\]
\[
7y = 35
\]
\[
y = \frac{35}{7} = 5
\]
Answer: \( y = 5 \)

14. \( 4y + 18 = 34 \)
\[
4y = 34 - 18 \quad \text{(Subtract 18 from both sides)}
\]
\[
4y = 16
\]
\[
y = \frac{16}{4} = 4
\]
Answer: \( y = 4 \)

15. \( 6y + 40 = 100 \)
\[
6y = 100 - 40 \quad \text{(Subtract 40 from both sides)}
\]
\[
6y = 60
\]
\[
y = \frac{60}{6} = 10
\]
Answer: \( y = 10 \)

16. \( 12y - 15 = 45 \)
\[
12y = 45 + 15 \quad \text{(Add 15 to both sides)}
\]
\[
12y = 60
\]
\[
y = \frac{60}{12} = 5
\]
Answer: \( y = 5 \)

17. \( 8y - 22 = 42 \)
\[
8y = 42 + 22 \quad \text{(Add 22 to both sides)}
\]
\[
8y = 64
\]
\[
y = \frac{64}{8} = 8
\]
Answer: \( y = 8 \)

18. \( 5y - 8 = 27 \)
\[
5y = 27 + 8 \quad \text{(Add 8 to both sides)}
\]
\[
5y = 35
\]
\[
y = \frac{35}{5} = 7
\]
Answer: \( y = 7 \)

19. \( 11y - 13 = 53 \)
\[
11y = 53 + 13 \quad \text{(Add 13 to both sides)}
\]
\[
11y = 66
\]
\[
y = \frac{66}{11} = 6
\]
Answer: \( y = 6 \)

20. \( 3y - 24 = 0 \)
\[
3y = 24 \quad \text{(Add 24 to both sides)}
\]
\[
y = \frac{24}{3} = 8
\]
Answer: \( y = 8 \)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1. y = 9 & 11. y = 10 \\
2. y = 9 & 12. y = 7 \\
3. y = 11 & 13. y = 5 \\
4. y = 12 & 14. y = 4 \\
5. y = 6 & 15. y = 10 \\
6. y = 5 & 16. y = 5 \\
7. y = 16 & 17. y = 8 \\
8. y = 8 & 18. y = 7 \\
9. y = 8 & 19. y = 6 \\
10. y = 9 & 20. y = 8 \\
\end{array}
}
\]