Algebraic Expressions & Identities (Addition & Subtraction ... - Free Printable

Problem: Add the following algebraic expressions

We will solve each problem step by step, combining like terms where applicable.

---

#### 1. \( 3a^2b, -4a^2b, 9a^2b \)

- All terms are like terms because they all have the same variables \( a^2b \).
- Combine the coefficients:
\[
3 + (-4) + 9 = 3 - 4 + 9 = 8
\]
- The result is:
\[
8a^2b
\]

Answer: \( 8a^2b \)

---

#### 2. \( \frac{2}{3}a, \frac{3}{5}a, -\frac{6}{5}a \)

- All terms are like terms because they all have the same variable \( a \).
- Combine the coefficients:
\[
\frac{2}{3} + \frac{3}{5} + \left(-\frac{6}{5}\right)
\]
- Find a common denominator for the fractions. The least common denominator (LCD) of 3 and 5 is 15.
\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
\]
\[
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
\]
\[
-\frac{6}{5} = -\frac{6 \times 3}{5 \times 3} = -\frac{18}{15}
\]
- Add the fractions:
\[
\frac{10}{15} + \frac{9}{15} - \frac{18}{15} = \frac{10 + 9 - 18}{15} = \frac{1}{15}
\]
- The result is:
\[
\frac{1}{15}a
\]

Answer: \( \frac{1}{15}a \)

---

#### 3. \( 4xy^2, -7x^2y, 12x^2y, -6xy^2, -3x^2y + 5xy^2 \)

- Group like terms:
- Terms with \( xy^2 \): \( 4xy^2, -6xy^2, 5xy^2 \)
- Terms with \( x^2y \): \( -7x^2y, 12x^2y, -3x^2y \)
- Combine the coefficients for \( xy^2 \):
\[
4 + (-6) + 5 = 4 - 6 + 5 = 3
\]
- Combine the coefficients for \( x^2y \):
\[
-7 + 12 + (-3) = -7 + 12 - 3 = 2
\]
- The result is:
\[
3xy^2 + 2x^2y
\]

Answer: \( 3xy^2 + 2x^2y \)

---

#### 4. \( \frac{3}{2}a - \frac{5}{4}b + \frac{2}{5}c, \frac{2}{3}a - \frac{7}{2}b + \frac{7}{2}c, \frac{5}{3}a + \frac{5}{2}b - \frac{5}{4}c \)

- Group like terms:
- Terms with \( a \): \( \frac{3}{2}a, \frac{2}{3}a, \frac{5}{3}a \)
- Terms with \( b \): \( -\frac{5}{4}b, -\frac{7}{2}b, \frac{5}{2}b \)
- Terms with \( c \): \( \frac{2}{5}c, \frac{7}{2}c, -\frac{5}{4}c \)
- Combine the coefficients for \( a \):
\[
\frac{3}{2} + \frac{2}{3} + \frac{5}{3}
\]
- Find a common denominator (LCD = 6):
\[
\frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}
\]
\[
\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
\]
\[
\frac{5}{3} = \frac{5 \times 2}{3 \times 2} = \frac{10}{6}
\]
- Add the fractions:
\[
\frac{9}{6} + \frac{4}{6} + \frac{10}{6} = \frac{9 + 4 + 10}{6} = \frac{23}{6}
\]
- Combine the coefficients for \( b \):
\[
-\frac{5}{4} - \frac{7}{2} + \frac{5}{2}
\]
- Find a common denominator (LCD = 4):
\[
-\frac{5}{4} = -\frac{5}{4}
\]
\[
-\frac{7}{2} = -\frac{7 \times 2}{2 \times 2} = -\frac{14}{4}
\]
\[
\frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4}
\]
- Add the fractions:
\[
-\frac{5}{4} - \frac{14}{4} + \frac{10}{4} = \frac{-5 - 14 + 10}{4} = \frac{-9}{4}
\]
- Combine the coefficients for \( c \):
\[
\frac{2}{5} + \frac{7}{2} - \frac{5}{4}
\]
- Find a common denominator (LCD = 20):
\[
\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}
\]
\[
\frac{7}{2} = \frac{7 \times 10}{2 \times 10} = \frac{70}{20}
\]
\[
-\frac{5}{4} = -\frac{5 \times 5}{4 \times 5} = -\frac{25}{20}
\]
- Add the fractions:
\[
\frac{8}{20} + \frac{70}{20} - \frac{25}{20} = \frac{8 + 70 - 25}{20} = \frac{53}{20}
\]
- The result is:
\[
\frac{23}{6}a - \frac{9}{4}b + \frac{53}{20}c
\]

Answer: \( \frac{23}{6}a - \frac{9}{4}b + \frac{53}{20}c \)

---

#### 5. \( \frac{11}{2}xy + \frac{12}{5}y + \frac{13}{7}x, -\frac{11}{2}xy - \frac{12}{5}x - \frac{137}{xy} \)

- Group like terms:
- Terms with \( xy \): \( \frac{11}{2}xy, -\frac{11}{2}xy \)
- Terms with \( y \): \( \frac{12}{5}y \)
- Terms with \( x \): \( \frac{13}{7}x, -\frac{12}{5}x \)
- Terms with \( \frac{1}{xy} \): \( -\frac{137}{xy} \)
- Combine the coefficients for \( xy \):
\[
\frac{11}{2} + \left(-\frac{11}{2}\right) = 0
\]
- Combine the coefficients for \( y \):
\[
\frac{12}{5}
\]
- Combine the coefficients for \( x \):
\[
\frac{13}{7} + \left(-\frac{12}{5}\right)
\]
- Find a common denominator (LCD = 35):
\[
\frac{13}{7} = \frac{13 \times 5}{7 \times 5} = \frac{65}{35}
\]
\[
-\frac{12}{5} = -\frac{12 \times 7}{5 \times 7} = -\frac{84}{35}
\]
- Add the fractions:
\[
\frac{65}{35} + \left(-\frac{84}{35}\right) = \frac{65 - 84}{35} = -\frac{19}{35}
\]
- The term \( -\frac{137}{xy} \) remains as it is because there are no other like terms.
- The result is:
\[
\frac{12}{5}y - \frac{19}{35}x - \frac{137}{xy}
\]

Answer: \( \frac{12}{5}y - \frac{19}{35}x - \frac{137}{xy} \)

---

#### 6. \( \frac{7}{2}x^3 - \frac{1}{2}x^2 + \frac{5}{3}, \frac{3}{2}x^3 + \frac{7}{4}x^2 - x + \frac{1}{3}, \frac{3}{2}x^2 - \frac{5}{2}x - 2 \)

- Group like terms:
- Terms with \( x^3 \): \( \frac{7}{2}x^3, \frac{3}{2}x^3 \)
- Terms with \( x^2 \): \( -\frac{1}{2}x^2, \frac{7}{4}x^2, \frac{3}{2}x^2 \)
- Terms with \( x \): \( -x, -\frac{5}{2}x \)
- Constant terms: \( \frac{5}{3}, \frac{1}{3}, -2 \)
- Combine the coefficients for \( x^3 \):
\[
\frac{7}{2} + \frac{3}{2} = \frac{10}{2} = 5
\]
- Combine the coefficients for \( x^2 \):
\[
-\frac{1}{2} + \frac{7}{4} + \frac{3}{2}
\]
- Find a common denominator (LCD = 4):
\[
-\frac{1}{2} = -\frac{1 \times 2}{2 \times 2} = -\frac{2}{4}
\]
\[
\frac{7}{4} = \frac{7}{4}
\]
\[
\frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}
\]
- Add the fractions:
\[
-\frac{2}{4} + \frac{7}{4} + \frac{6}{4} = \frac{-2 + 7 + 6}{4} = \frac{11}{4}
\]
- Combine the coefficients for \( x \):
\[
-1 - \frac{5}{2}
\]
- Find a common denominator (LCD = 2):
\[
-1 = -\frac{2}{2}
\]
\[
-\frac{2}{2} - \frac{5}{2} = \frac{-2 - 5}{2} = -\frac{7}{2}
\]
- Combine the constant terms:
\[
\frac{5}{3} + \frac{1}{3} - 2
\]
- Find a common denominator (LCD = 3):
\[
\frac{5}{3} = \frac{5}{3}
\]
\[
\frac{1}{3} = \frac{1}{3}
\]
\[
-2 = -\frac{6}{3}
\]
- Add the fractions:
\[
\frac{5}{3} + \frac{1}{3} - \frac{6}{3} = \frac{5 + 1 - 6}{3} = 0
\]
- The result is:
\[
5x^3 + \frac{11}{4}x^2 - \frac{7}{2}x
\]

Answer: \( 5x^3 + \frac{11}{4}x^2 - \frac{7}{2}x \)

---

#### 7. \( 6a + 8b - 7c, 2b + c - 4a, a - 3b - 2c \)

- Group like terms:
- Terms with \( a \): \( 6a, -4a, a \)
- Terms with \( b \): \( 8b, 2b, -3b \)
- Terms with \( c \): \( -7c, c, -2c \)
- Combine the coefficients for \( a \):
\[
6 + (-4) + 1 = 6 - 4 + 1 = 3
\]
- Combine the coefficients for \( b \):
\[
8 + 2 + (-3) = 8 + 2 - 3 = 7
\]
- Combine the coefficients for \( c \):
\[
-7 + 1 + (-2) = -7 + 1 - 2 = -8
\]
- The result is:
\[
3a + 7b - 8c
\]

Answer: \( 3a + 7b - 8c \)

---

#### 8. \( 5x^2 + 7y - 8, 4y + 7 - 2x^2, 6 - 5y + 4x^2 \)

- Group like terms:
- Terms with \( x^2 \): \( 5x^2, -2x^2, 4x^2 \)
- Terms with \( y \): \( 7y, 4y, -5y \)
- Constant terms: \( -8, 7, 6 \)
- Combine the coefficients for \( x^2 \):
\[
5 + (-2) + 4 = 5 - 2 + 4 = 7
\]
- Combine the coefficients for \( y \):
\[
7 + 4 + (-5) = 7 + 4 - 5 = 6
\]
- Combine the constant terms:
\[
-8 + 7 + 6 = -8 + 7 + 6 = 5
\]
- The result is:
\[
7x^2 + 6y + 5
\]

Answer: \( 7x^2 + 6y + 5 \)

---

#### 9. \( 8x^2 - 5xy + 3y^2, 2xy - 6y^2 + 3x^2, y^2 + xy - 6x^2 \)

- Group like terms:
- Terms with \( x^2 \): \( 8x^2, 3x^2, -6x^2 \)
- Terms with \( xy \): \( -5xy, 2xy, xy \)
- Terms with \( y^2 \): \( 3y^2, -6y^2, y^2 \)
- Combine the coefficients for \( x^2 \):
\[
8 + 3 + (-6) = 8 + 3 - 6 = 5
\]
- Combine the coefficients for \( xy \):
\[
-5 + 2 + 1 = -5 + 2 + 1 = -2
\]
- Combine the coefficients for \( y^2 \):
\[
3 + (-6) + 1 = 3 - 6 + 1 = -2
\]
- The result is:
\[
5x^2 - 2xy - 2y^2
\]

Answer: \( 5x^2 - 2xy - 2y^2 \)

---

#### 10. \( 5x^3 - 2y^3 \) and \( 7x^3 - 3y^3 \)

- Group like terms:
- Terms with \( x^3 \): \( 5x^3, 7x^3 \)
- Terms with \( y^3 \): \( -2y^3, -3y^3 \)
- Combine the coefficients for \( x^3 \):
\[
5 + 7 = 12
\]
- Combine the coefficients for \( y^3 \):
\[
-2 + (-3) = -2 - 3 = -5
\]
- The result is:
\[
12x^3 - 5y^3
\]

Answer: \( 12x^3 - 5y^3 \)

---

Final Answers:

1. \( 8a^2b \)
2. \( \frac{1}{15}a \)
3. \( 3xy^2 + 2x^2y \)
4. \( \frac{23}{6}a - \frac{9}{4}b + \frac{53}{20}c \)
5. \( \frac{12}{5}y - \frac{19}{35}x - \frac{137}{xy} \)
6. \( 5x^3 + \frac{11}{4}x^2 - \frac{7}{2}x \)
7. \( 3a + 7b - 8c \)
8. \( 7x^2 + 6y + 5 \)
9. \( 5x^2 - 2xy - 2y^2 \)
10. \( 12x^3 - 5y^3 \)

\boxed{
\begin{aligned}
1. & \ 8a^2b \\
2. & \ \frac{1}{15}a \\
3. & \ 3xy^2 + 2x^2y \\
4. & \ \frac{23}{6}a - \frac{9}{4}b + \frac{53}{20}c \\
5. & \ \frac{12}{5}y - \frac{19}{35}x - \frac{137}{xy} \\
6. & \ 5x^3 + \frac{11}{4}x^2 - \frac{7}{2}x \\
7. & \ 3a + 7b - 8c \\
8. & \ 7x^2 + 6y + 5 \\
9. & \ 5x^2 - 2xy - 2y^2 \\
10. & \ 12x^3 - 5y^3
\end{aligned}
}