Single-Term & Linear Expressions

• Problem 1: Add: $4a + 7a$

• Problem 2: Add: $5x + 3 + 2x + 9$

• Problem 8: Add: $6m – 4n + 3, -2m + 5n – 7$

• Problem 9: Add: $2p + 3q, -5p + 6q, 4p – q$

• Problem 10: Add: $4u + 3v, -2u – 5v, 7u + v$

Expressions with Exponents

• Problem 3: Add: $x^2 + 2y^2 + 3xy, 4x^2 – y^2 – xy$

Expressions with Fractions

• Problem 4: Add: $\frac{a}{3} + \frac{2a}{5}$

• Problem 5: Add: $\frac{3}{4}x^2 + \frac{5}{6}x^2 + \frac{1}{3}x^2$

Vertical Method Practice

• Problem 6: Using vertical method, add: $3a + 5b, 7a + 2b$

• Problem 7: Using vertical method, add: $2x^2 + 3y^2 + xy, 5x^2 – y^2 – 2xy$

Algebraic Expressions: Solutions & Explanations

1. Add: $4a + 7a$

• Explanation: Since both terms have the exact same variable ($a$), they are “like terms.” To add them, you simply add the numerical coefficients ($4 + 7$) and keep the variable.

• Result: $11a$

2. Add: $5x + 3 + 2x + 9$

• Explanation: Group the terms that are alike. Combine the $x$ terms ($5x + 2x$) and the constant numbers ($3 + 9$) separately.

• Result: $7x + 12$

3. Add: $x^2 + 2y^2 + 3xy, 4x^2 – y^2 – xy$

• Explanation: Look for terms with matching variables and exponents. Combine $1x^2$ with $4x^2$, $2y^2$ with $-1y^2$, and $3xy$ with $-1xy$.

• Result: $5x^2 + y^2 + 2xy$

4. Add: $\frac{a}{3} + \frac{2a}{5}$

• Explanation: To add algebraic fractions, you must find a common denominator. For $3$ and $5$, the common denominator is $15$. Convert the fractions: $\frac{5a}{15} + \frac{6a}{15}$.

• Result: $\frac{11a}{15}$

5. Add: $\frac{3}{4}x^2 + \frac{5}{6}x^2 + \frac{1}{3}x^2$

• Explanation: All terms are like terms ($x^2$). Find a common denominator for the coefficients ($4, 6,$ and $3$), which is $12$. This gives $\frac{9}{12} + \frac{10}{12} + \frac{4}{12}$.

• Result: $\frac{23}{12}x^2$ (or $1\frac{11}{12}x^2$)

6. Using vertical method, add: $3a + 5b, 7a + 2b$

• Explanation: Align the variables in columns so that $a$ is over $a$ and $b$ is over $b$. Add straight down each column.

• Result: $10a + 7b$

7. Using vertical method, add: $2x^2 + 3y^2 + xy, 5x^2 – y^2 – 2xy$

• Explanation: Stack the expressions so $x^2$, $y^2$, and $xy$ terms are aligned. $3y^2 + (-y^2)$ becomes $2y^2$, and $xy + (-2xy)$ becomes $-xy$.

• Result: $7x^2 + 2y^2 – xy$

8. Add: $6m – 4n + 3, -2m + 5n – 7$

• Explanation: Combine the $m$ terms ($6 – 2$), the $n$ terms ($-4 + 5$), and the constants ($3 – 7$). Be careful with the negative signs.

• Result: $4m + n – 4$

9. Add: $2p + 3q, -5p + 6q, 4p – q$

• Explanation: Add the three $p$ coefficients ($2 – 5 + 4$) and the three $q$ coefficients ($3 + 6 – 1$).

• Result: $p + 8q$

10. Add: $4u + 3v, -2u – 5v, 7u + v$

• Explanation: Total the $u$ terms ($4 – 2 + 7$) and the $v$ terms ($3 – 5 + 1$).

• Result: $9u – v$