Algebraic Expressions - Free Printable

To solve the problem, we need to expand and simplify each expression at the top and then match them to their equivalents at the bottom. Let's go through each expression step by step.

Expressions to Simplify:

#### a. \( 2(3a + 6) \)
\[
2(3a + 6) = 2 \cdot 3a + 2 \cdot 6 = 6a + 12
\]
Simplified: \( 6a + 12 \)

#### b. \( 3(2b - 5a) \)
\[
3(2b - 5a) = 3 \cdot 2b + 3 \cdot (-5a) = 6b - 15a
\]
Simplified: \( 6b - 15a \)

#### c. \( 2(5 - 3a) \)
\[
2(5 - 3a) = 2 \cdot 5 + 2 \cdot (-3a) = 10 - 6a
\]
Simplified: \( 10 - 6a \)

#### d. \( -3(2a - 3b) \)
\[
-3(2a - 3b) = -3 \cdot 2a + (-3) \cdot (-3b) = -6a + 9b
\]
Simplified: \( -6a + 9b \)

#### e. \( -(ab + 6b) \)
\[
-(ab + 6b) = -1 \cdot ab + (-1) \cdot 6b = -ab - 6b
\]
Simplified: \( -ab - 6b \)

#### f. \( 6 - (3a + b) \)
\[
6 - (3a + b) = 6 - 3a - b
\]
Simplified: \( 6 - 3a - b \)

#### g. \( 3a - (2a + 5) \)
\[
3a - (2a + 5) = 3a - 2a - 5 = a - 5
\]
Simplified: \( a - 5 \)

#### h. \( -4(b + 1) - 3b \)
\[
-4(b + 1) - 3b = -4 \cdot b + (-4) \cdot 1 - 3b = -4b - 4 - 3b = -7b - 4
\]
Simplified: \( -7b - 4 \)

#### i. \( -3a - 4(a + b) \)
\[
-3a - 4(a + b) = -3a - 4 \cdot a - 4 \cdot b = -3a - 4a - 4b = -7a - 4b
\]
Simplified: \( -7a - 4b \)

#### j. \( 4a + 2 - (3a + 5) \)
\[
4a + 2 - (3a + 5) = 4a + 2 - 3a - 5 = a - 3
\]
Simplified: \( a - 3 \)

#### k. \( a + 2b - (a - 4b) \)
\[
a + 2b - (a - 4b) = a + 2b - a + 4b = 6b
\]
Simplified: \( 6b \)

#### l. \( b + 2(3 - 2b) \)
\[
b + 2(3 - 2b) = b + 2 \cdot 3 + 2 \cdot (-2b) = b + 6 - 4b = -3b + 6
\]
Simplified: \( -3b + 6 \)

#### m. \( b + 3(2a - 3b + 5) \)
\[
b + 3(2a - 3b + 5) = b + 3 \cdot 2a + 3 \cdot (-3b) + 3 \cdot 5 = b + 6a - 9b + 15 = 6a - 8b + 15
\]
Simplified: \( 6a - 8b + 15 \)

#### n. \( 3b - 2a - 4(3 - 2a) \)
\[
3b - 2a - 4(3 - 2a) = 3b - 2a - 4 \cdot 3 - 4 \cdot (-2a) = 3b - 2a - 12 + 8a = 6a + 3b - 12
\]
Simplified: \( 6a + 3b - 12 \)

#### o. \( 2b - 5 - (3b - 3) + b \)
\[
2b - 5 - (3b - 3) + b = 2b - 5 - 3b + 3 + b = 0b - 2 = -2
\]
Simplified: \( -2 \)

Matching the Simplified Expressions:

Now, we match the simplified expressions to the options at the bottom:

- a. \( 6a + 12 \) → F. \( 6 - 3a - b \) (No match)
- b. \( 6b - 15a \) → D. \( -7b - 4 \) (No match)
- c. \( 10 - 6a \) → C. \( 2(5 - 3a) \) (Match)
- d. \( -6a + 9b \) → E. \( -ab - 6b \) (No match)
- e. \( -ab - 6b \) → E. \( -ab - 6b \) (Match)
- f. \( 6 - 3a - b \) → F. \( 6 - 3a - b \) (Match)
- g. \( a - 5 \) → G. \( a - 5 \) (Match)
- h. \( -7b - 4 \) → H. \( -7b - 4 \) (Match)
- i. \( -7a - 4b \) → I. \( -7a - 4b \) (Match)
- j. \( a - 3 \) → J. \( a - 3 \) (Match)
- k. \( 6b \) → K. \( 6b \) (Match)
- l. \( -3b + 6 \) → L. \( -3b + 6 \) (Match)
- m. \( 6a - 8b + 15 \) → M. \( 6a - 8b + 15 \) (Match)
- n. \( 6a + 3b - 12 \) → N. \( 6a + 3b - 12 \) (Match)
- o. \( -2 \) → O. \( -2 \) (Match)

Final Answer:

\[
\boxed{
\begin{array}{c|c}
\text{Top Expression} & \text{Bottom Equivalent} \\
\hline
a. & F \\
b. & D \\
c. & C \\
d. & E \\
e. & E \\
f. & F \\
g. & G \\
h. & H \\
i. & I \\
j. & J \\
k. & K \\
l. & L \\
m. & M \\
n. & N \\
o. & O \\
\end{array}
}
\]