Evaluating Algebraic Expressions (Worksheets) - Free Printable

Problem: Evaluate the given expressions for the specified values of variables.

#### Step 1: Evaluate the expressions for \( y = -3 \)

1. Expression: \( \frac{2(y + 6)}{3} \)
- Substitute \( y = -3 \):
\[
\frac{2((-3) + 6)}{3} = \frac{2(3)}{3} = \frac{6}{3} = 2
\]
- Answer: \( 2 \)

2. Expression: \( -2|y| \)
- Substitute \( y = -3 \):
\[
-2|y| = -2|-3| = -2 \cdot 3 = -6
\]
- Answer: \( -6 \)

3. Expression: \( y(|y| - 2) \)
- Substitute \( y = -3 \):
\[
y(|y| - 2) = (-3)(|-3| - 2) = (-3)(3 - 2) = (-3)(1) = -3
\]
- Answer: \( -3 \)

4. Expression: \( \frac{2|y|}{6} \)
- Substitute \( y = -3 \):
\[
\frac{2|y|}{6} = \frac{2|-3|}{6} = \frac{2 \cdot 3}{6} = \frac{6}{6} = 1
\]
- Answer: \( 1 \)

5. Expression: \( \frac{9y + 5y}{-3} \)
- Simplify the numerator first:
\[
9y + 5y = 14y
\]
- Substitute \( y = -3 \):
\[
\frac{14y}{-3} = \frac{14(-3)}{-3} = \frac{-42}{-3} = 14
\]
- Answer: \( 14 \)

6. Expression: \( \frac{5(y + 6)}{2} \)
- Substitute \( y = -3 \):
\[
\frac{5((-3) + 6)}{2} = \frac{5(3)}{2} = \frac{15}{2}
\]
- Answer: \( \frac{15}{2} \)

7. Expression: \( \frac{y^3}{3} \)
- Substitute \( y = -3 \):
\[
\frac{y^3}{3} = \frac{(-3)^3}{3} = \frac{-27}{3} = -9
\]
- Answer: \( -9 \)

8. Expression: \( \frac{18}{y^2} \)
- Substitute \( y = -3 \):
\[
\frac{18}{y^2} = \frac{18}{(-3)^2} = \frac{18}{9} = 2
\]
- Answer: \( 2 \)

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#### Step 2: Evaluate the expressions for \( x = -2 \) and \( c = 8 \)

9. Expression: \( -2|x - c| \)
- Substitute \( x = -2 \) and \( c = 8 \):
\[
-2|x - c| = -2|(-2) - 8| = -2|-10| = -2 \cdot 10 = -20
\]
- Answer: \( -20 \)

10. Expression: \( \frac{9x + 8}{2} \)
- Substitute \( x = -2 \):
\[
\frac{9x + 8}{2} = \frac{9(-2) + 8}{2} = \frac{-18 + 8}{2} = \frac{-10}{2} = -5
\]
- Answer: \( -5 \)

11. Expression: \( \frac{-3x - 8}{2} \)
- Substitute \( x = -2 \):
\[
\frac{-3x - 8}{2} = \frac{-3(-2) - 8}{2} = \frac{6 - 8}{2} = \frac{-2}{2} = -1
\]
- Answer: \( -1 \)

12. Expression: \( \frac{-c(x + 4)}{8} \)
- Substitute \( x = -2 \) and \( c = 8 \):
\[
\frac{-c(x + 4)}{8} = \frac{-8((-2) + 4)}{8} = \frac{-8(2)}{8} = \frac{-16}{8} = -2
\]
- Answer: \( -2 \)

13. Expression: \( \frac{x^3}{2} + 4 \)
- Substitute \( x = -2 \):
\[
\frac{x^3}{2} + 4 = \frac{(-2)^3}{2} + 4 = \frac{-8}{2} + 4 = -4 + 4 = 0
\]
- Answer: \( 0 \)

14. Expression: \( \frac{3(c + x^2)}{12} \)
- Substitute \( x = -2 \) and \( c = 8 \):
\[
\frac{3(c + x^2)}{12} = \frac{3(8 + (-2)^2)}{12} = \frac{3(8 + 4)}{12} = \frac{3 \cdot 12}{12} = \frac{36}{12} = 3
\]
- Answer: \( 3 \)

15. Expression: \( \frac{\sqrt{2}}{x - c} \)
- Substitute \( x = -2 \) and \( c = 8 \):
\[
\frac{\sqrt{2}}{x - c} = \frac{\sqrt{2}}{(-2) - 8} = \frac{\sqrt{2}}{-10} = -\frac{\sqrt{2}}{10}
\]
- Answer: \( -\frac{\sqrt{2}}{10} \)

16. Expression: \( \frac{15x - c}{-10} \)
- Substitute \( x = -2 \) and \( c = 8 \):
\[
\frac{15x - c}{-10} = \frac{15(-2) - 8}{-10} = \frac{-30 - 8}{-10} = \frac{-38}{-10} = \frac{38}{10} = \frac{19}{5}
\]
- Answer: \( \frac{19}{5} \)

---

#### Step 3: Evaluate the expressions for \( d = 6 \) and \( a = -4 \)

17. Expression: \( \frac{d(a)}{-5} \)
- Substitute \( d = 6 \) and \( a = -4 \):
\[
\frac{d(a)}{-5} = \frac{6(-4)}{-5} = \frac{-24}{-5} = \frac{24}{5}
\]
- Answer: \( \frac{24}{5} \)

18. Expression: \( \frac{a^3 - d^3}{4} \)
- Substitute \( d = 6 \) and \( a = -4 \):
\[
a^3 = (-4)^3 = -64, \quad d^3 = 6^3 = 216
\]
\[
\frac{a^3 - d^3}{4} = \frac{-64 - 216}{4} = \frac{-280}{4} = -70
\]
- Answer: \( -70 \)

19. Expression: \( \frac{d(a + 3)}{2} \)
- Substitute \( d = 6 \) and \( a = -4 \):
\[
\frac{d(a + 3)}{2} = \frac{6((-4) + 3)}{2} = \frac{6(-1)}{2} = \frac{-6}{2} = -3
\]
- Answer: \( -3 \)

20. Expression: \( \frac{a(d^2 - 5)}{6} \)
- Substitute \( d = 6 \) and \( a = -4 \):
\[
d^2 = 6^2 = 36
\]
\[
\frac{a(d^2 - 5)}{6} = \frac{-4(36 - 5)}{6} = \frac{-4 \cdot 31}{6} = \frac{-124}{6} = -\frac{62}{3}
\]
- Answer: \( -\frac{62}{3} \)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 2 \\
2. & -6 \\
3. & -3 \\
4. & 1 \\
5. & 14 \\
6. & \frac{15}{2} \\
7. & -9 \\
8. & 2 \\
9. & -20 \\
10. & -5 \\
11. & -1 \\
12. & -2 \\
13. & 0 \\
14. & 3 \\
15. & -\frac{\sqrt{2}}{10} \\
16. & \frac{19}{5} \\
17. & \frac{24}{5} \\
18. & -70 \\
19. & -3 \\
20. & -\frac{62}{3}
\end{array}
}
\]