Algebraic expressions worksheet from Chapter 12, featuring definitions, examples, and practice problems on like and unlike terms, coefficients, and polynomial operations.

Problem: Solve the given algebraic expression problems from Chapter 12.

#### Points to Remember:
1. Algebraic Expression: Formed from variables and constants.
2. Like Terms: Terms with the same algebraic factors.
3. Unlike Terms: Terms with different algebraic factors.
4. Polynomial: An expression with one or more terms where the power of variables is a positive integer.

---

Questions and Solutions:

#### Question 1: Add the like terms:
$$
\frac{2}{3}x^2y, \quad -\frac{3}{7}x^2y, \quad \frac{2}{7}x^2y^3
$$

- Step 1: Identify like terms.
- The terms $\frac{2}{3}x^2y$ and $-\frac{3}{7}x^2y$ are like terms because they have the same algebraic factor $x^2y$.
- The term $\frac{2}{7}x^2y^3$ is not a like term with the others because it has a different algebraic factor ($x^2y^3$).

- Step 2: Add the like terms $\frac{2}{3}x^2y$ and $-\frac{3}{7}x^2y$.
$$
\frac{2}{3}x^2y + \left(-\frac{3}{7}x^2y\right) = \left(\frac{2}{3} - \frac{3}{7}\right)x^2y
$$
- Find a common denominator for $\frac{2}{3}$ and $\frac{3}{7}$:
$$
\text{Common denominator} = 21
$$
$$
\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}, \quad \frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21}
$$
$$
\frac{2}{3} - \frac{3}{7} = \frac{14}{21} - \frac{9}{21} = \frac{5}{21}
$$

- Therefore:
$$
\frac{2}{3}x^2y + \left(-\frac{3}{7}x^2y\right) = \frac{5}{21}x^2y
$$

- Step 3: Combine the results.
$$
\frac{2}{3}x^2y + \left(-\frac{3}{7}x^2y\right) + \frac{2}{7}x^2y^3 = \frac{5}{21}x^2y + \frac{2}{7}x^2y^3
$$

- Final Answer:
$$
\boxed{\frac{5}{21}x^2y + \frac{2}{7}x^2y^3}
$$

---

#### Question 2: Add the unlike terms:
$$
8x, \quad -\frac{4}{3}x, \quad \frac{2}{3}y, \quad 3p
$$

- Step 1: Identify that all terms are unlike terms because they have different algebraic factors.
- $8x$: Factor is $x$.
- $-\frac{4}{3}x$: Factor is $x$.
- $\frac{2}{3}y$: Factor is $y$.
- $3p$: Factor is $p$.

- Step 2: Since these are unlike terms, they cannot be combined further.
$$
8x - \frac{4}{3}x + \frac{2}{3}y + 3p
$$

- Final Answer:
$$
\boxed{8x - \frac{4}{3}x + \frac{2}{3}y + 3p}
$$

---

#### Question 3: Add the unlike terms:
$$
4x^2, \quad 2y, \quad 3z, \quad 4x^2
$$

- Step 1: Identify like terms.
- $4x^2$ and $4x^2$ are like terms because they have the same algebraic factor $x^2$.
- $2y$, $3z$, and $4x^2$ are unlike terms.

- Step 2: Add the like terms $4x^2$ and $4x^2$.
$$
4x^2 + 4x^2 = 8x^2
$$

- Step 3: Combine the results.
$$
8x^2 + 2y + 3z
$$

- Final Answer:
$$
\boxed{8x^2 + 2y + 3z}
$$

---

#### Question 4: Add the terms:
$$
x^2 \quad \text{and} \quad -3y^2
$$

- Step 1: Identify that these are unlike terms because they have different algebraic factors.
- $x^2$: Factor is $x^2$.
- $-3y^2$: Factor is $y^2$.

- Step 2: Since these are unlike terms, they cannot be combined further.
$$
x^2 - 3y^2
$$

- Final Answer:
$$
\boxed{x^2 - 3y^2}
$$

---

#### Question 5: What is the numerical coefficient of the algebraic expression $13 - y^2$?

- Step 1: Identify the terms in the expression.
- $13$: Constant term.
- $-y^2$: Term with variable $y^2$.

- Step 2: The numerical coefficient of $-y^2$ is the constant multiplier of $y^2$, which is $-1$.

- Final Answer:
$$
\boxed{-1}
$$

---

#### Question 6: Add all the terms:
$$
12x, \quad 12, \quad -25x, \quad -25y, \quad 1, \quad x, \quad 12y, \quad y, \quad -25
$$

- Step 1: Group like terms.
- Terms involving $x$: $12x$, $-25x$, $x$.
- Terms involving $y$: $-25y$, $12y$, $y$.
- Constant terms: $12$, $1$, $-25$.

- Step 2: Add the like terms.
- For $x$: $12x - 25x + x = (12 - 25 + 1)x = -12x$.
- For $y$: $-25y + 12y + y = (-25 + 12 + 1)y = -12y$.
- For constants: $12 + 1 - 25 = -12$.

- Step 3: Combine the results.
$$
-12x - 12y - 12
$$

- Final Answer:
$$
\boxed{-12x - 12y - 12}
$$

---

#### Question 7: Write the coefficient of $y^2$ in the algebraic expression:
$$
2x^2y - 15xy^2 + 7y^2
$$

- Step 1: Identify the term containing $y^2$.
- The term is $7y^2$.

- Step 2: The coefficient of $y^2$ is the constant multiplier of $y^2$, which is $7$.

- Final Answer:
$$
\boxed{7}
$$

---

#### Question 8: Add the algebraic expressions:
$$
3x + 11 \quad \text{and} \quad -2x + y
$$

- Step 1: Combine like terms.
- Terms involving $x$: $3x$ and $-2x$.
- Terms involving $y$: $y$ (no other $y$-terms).
- Constant terms: $11$ (no other constant terms).

- Step 2: Add the like terms.
- For $x$: $3x - 2x = x$.
- For $y$: $y$ (remains as is).
- For constants: $11$ (remains as is).

- Step 3: Combine the results.
$$
x + y + 11
$$

- Final Answer:
$$
\boxed{x + y + 11}
$$

---

#### Question 9: Subtract $x + y$ from $\frac{3}{2}x - \frac{5}{2}y$.

- Step 1: Write the subtraction as an expression.
$$
\left(\frac{3}{2}x - \frac{5}{2}y\right) - (x + y)
$$

- Step 2: Distribute the negative sign.
$$
\frac{3}{2}x - \frac{5}{2}y - x - y
$$

- Step 3: Combine like terms.
- For $x$: $\frac{3}{2}x - x = \frac{3}{2}x - \frac{2}{2}x = \frac{1}{2}x$.
- For $y$: $-\frac{5}{2}y - y = -\frac{5}{2}y - \frac{2}{2}y = -\frac{7}{2}y$.

- Step 4: Combine the results.
$$
\frac{1}{2}x - \frac{7}{2}y
$$

- Final Answer:
$$
\boxed{\frac{1}{2}x - \frac{7}{2}y}
$$

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#### Question 10: Find the value of the following expression for $x = 2$:
$$
15x - 5x^2
$$

- Step 1: Substitute $x = 2$ into the expression.
$$
15(2) - 5(2)^2
$$

- Step 2: Simplify step by step.
- Calculate $15(2)$:
$$
15(2) = 30
$$
- Calculate $(2)^2$:
$$
(2)^2 = 4
$$
- Calculate $5(2)^2$:
$$
5(4) = 20
$$
- Subtract:
$$
30 - 20 = 10
$$

- Final Answer:
$$
\boxed{10}
$$

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Final Answers:

1. $\boxed{\frac{5}{21}x^2y + \frac{2}{7}x^2y^3}$
2. $\boxed{8x - \frac{4}{3}x + \frac{2}{3}y + 3p}$
3. $\boxed{8x^2 + 2y + 3z}$
4. $\boxed{x^2 - 3y^2}$
5. $\boxed{-1}$
6. $\boxed{-12x - 12y - 12}$
7. $\boxed{7}$
8. $\boxed{x + y + 11}$
9. $\boxed{\frac{1}{2}x - \frac{7}{2}y}$
10. $\boxed{10}$