Mathematics 6 worksheet featuring algebra exercises on like terms, simplification, and evaluation.

Worksheet: Mathematics 6

#### Part A: List down all the like terms in each set.

Like terms are terms that have the same variables raised to the same powers. We will group these terms together.

1. Set: \( 5x, 3y, x, y^2, 6z \)
- Like terms: \( 5x \) and \( x \) (both contain \( x \))
- No other like terms.

2. Set: \( a^2, 2b, 7a, b^2, 7b \)
- Like terms: \( 2b \) and \( 7b \) (both contain \( b \))
- No other like terms.

3. Set: \( 10z, 6x, 8y, z, 2a \)
- Like terms: \( 10z \) and \( z \) (both contain \( z \))
- No other like terms.

4. Set: \( x^2y, 8yx, 9xy^2, -4xy, x^2y^2 \)
- Like terms: \( x^2y \) and \( 8yx \) (both are \( xy^2 \))
- No other like terms.

5. Set: \( 7c^2d, 11d^2c, 2dc, c^2d^2, 3d^2c^2 \)
- Like terms: \( 7c^2d \) and \( 11d^2c \) (both are \( cd^2 \))
- No other like terms.

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#### Part B: Simplify the following algebraic expressions by combining the like terms.

1. Expression: \( 5a + 6a - 7a \)
- Combine like terms: \( (5 + 6 - 7)a = 4a \)
- Answer: \( 4a \)

2. Expression: \( 4d - 2d + 8d - c \)
- Combine like terms for \( d \): \( (4 - 2 + 8)d = 10d \)
- The term \( -c \) remains as it is.
- Answer: \( 10d - c \)

3. Expression: \( 12x - 5x + 5y - 4y \)
- Combine like terms for \( x \): \( (12 - 5)x = 7x \)
- Combine like terms for \( y \): \( (5 - 4)y = y \)
- Answer: \( 7x + y \)

4. Expression: \( 9x^2 + 3z - 6x^2 + 4z - 5z \)
- Combine like terms for \( x^2 \): \( (9 - 6)x^2 = 3x^2 \)
- Combine like terms for \( z \): \( (3 + 4 - 5)z = 2z \)
- Answer: \( 3x^2 + 2z \)

5. Expression: \( ab + 3xy + 8ab - 4ab + 7xy - 2xy \)
- Combine like terms for \( ab \): \( (1 + 8 - 4)ab = 5ab \)
- Combine like terms for \( xy \): \( (3 + 7 - 2)xy = 8xy \)
- Answer: \( 5ab + 8xy \)

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#### Part C: Evaluate each expression when \( a = 2 \), \( b = 3 \), and \( c = 4 \).

1. Expression: \( 10a \)
- Substitute \( a = 2 \): \( 10(2) = 20 \)
- Answer: \( 20 \)

2. Expression: \( 6b + c \)
- Substitute \( b = 3 \) and \( c = 4 \): \( 6(3) + 4 = 18 + 4 = 22 \)
- Answer: \( 22 \)

3. Expression: \( \frac{b^2 - a^2}{5} \)
- Substitute \( a = 2 \) and \( b = 3 \):
\[
b^2 = 3^2 = 9, \quad a^2 = 2^2 = 4
\]
\[
b^2 - a^2 = 9 - 4 = 5
\]
\[
\frac{b^2 - a^2}{5} = \frac{5}{5} = 1
\]
- Answer: \( 1 \)

4. Expression: \( (ab + c) - a^3 \)
- Substitute \( a = 2 \), \( b = 3 \), and \( c = 4 \):
\[
ab = 2 \cdot 3 = 6, \quad ab + c = 6 + 4 = 10
\]
\[
a^3 = 2^3 = 8
\]
\[
(ab + c) - a^3 = 10 - 8 = 2
\]
- Answer: \( 2 \)

5. Expression: \( \frac{5b^2 - (c + 1)}{a + b} \)
- Substitute \( a = 2 \), \( b = 3 \), and \( c = 4 \):
\[
b^2 = 3^2 = 9, \quad 5b^2 = 5 \cdot 9 = 45
\]
\[
c + 1 = 4 + 1 = 5
\]
\[
5b^2 - (c + 1) = 45 - 5 = 40
\]
\[
a + b = 2 + 3 = 5
\]
\[
\frac{5b^2 - (c + 1)}{a + b} = \frac{40}{5} = 8
\]
- Answer: \( 8 \)

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Final Answers:

- Part A:
1. \( 5x, x \)
2. \( 2b, 7b \)
3. \( 10z, z \)
4. \( x^2y, 8yx \)
5. \( 7c^2d, 11d^2c \)

- Part B:
1. \( 4a \)
2. \( 10d - c \)
3. \( 7x + y \)
4. \( 3x^2 + 2z \)
5. \( 5ab + 8xy \)

- Part C:
1. \( 20 \)
2. \( 22 \)
3. \( 1 \)
4. \( 2 \)
5. \( 8 \)

Final Boxed Answers:
\[
\boxed{
\begin{array}{l}
\text{Part A:} \\
1. \{5x, x\} \\
2. \{2b, 7b\} \\
3. \{10z, z\} \\
4. \{x^2y, 8yx\} \\
5. \{7c^2d, 11d^2c\} \\
\\
\text{Part B:} \\
1. 4a \\
2. 10d - c \\
3. 7x + y \\
4. 3x^2 + 2z \\
5. 5ab + 8xy \\
\\
\text{Part C:} \\
1. 20 \\
2. 22 \\
3. 1 \\
4. 2 \\
5. 8 \\
\end{array}
}
\]