Match equivalent algebraic expressions from two columns.

Let's solve this step by step. We are given expressions in Column 1 and need to match each one to an equivalent expression in Column 2.

We'll simplify each expression in Column 1 and see which one in Column 2 matches it.

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A. $ a(1 + 2 + 3) $

First, simplify inside the parentheses:
$$
1 + 2 + 3 = 6
$$
So,
$$
a(6) = 6a
$$

Now look for an expression in Column 2 that equals $6a$.
But none of the options directly say $6a$, so let’s check if any can be simplified to $6a$.

Wait — look at Option 4: $(2 + 3)a = 5a$ → no
Option 5: $a + 2a + 3a = (1+2+3)a = 6a$ ✔

So A matches with 5.

> ✔ A → 5

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B. $ 2(12 - 4) $

Simplify inside:
$$
12 - 4 = 8
\Rightarrow 2 \cdot 8 = 16
$$

Now look for an expression in Column 2 that equals 16.

Option 2: $12 \cdot 2 - 4 \cdot 2 = 24 - 8 = 16$ ✔

So B matches with 2

> ✔ B → 2

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C. $ 12a + 3b $

Look for a matching expression.

Option 3: $2(3a + 5b) = 6a + 10b$ → not matching
Option 1: $3(4a + b) = 12a + 3b$ ✔

Yes! That's exactly $12a + 3b$

So C matches with 1

> ✔ C → 1

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D. $ \frac{2}{3}(15a - 18) $

Distribute $ \frac{2}{3} $:

$$
\frac{2}{3} \cdot 15a = 10a \\
\frac{2}{3} \cdot (-18) = -12
$$

So total: $10a - 12$

Now look in Column 2: Option 6 is $10a - 12$ ✔

So D matches with 6

> ✔ D → 6

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E. $ 6a + 10b $

Check if any expression simplifies to this.

Option 3: $2(3a + 5b) = 6a + 10b$ ✔

So E matches with 3

> ✔ E → 3

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F. $ 0.4(5 - 2.5a) $

Distribute $0.4$:

$$
0.4 \cdot 5 = 2 \\
0.4 \cdot (-2.5a) = -1a = -a
$$

So result: $2 - a$

Now look in Column 2: Option 7 is $2 - a$ ✔

So F matches with 7

> ✔ F → 7

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G. $ 2a + 3a $

Combine like terms:
$$
2a + 3a = 5a
$$

Now look for an expression equal to $5a$

Option 4: $(2 + 3)a = 5a$ ✔

So G matches with 4

> ✔ G → 4

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Final Matching:

| Column 1 | Matches | Column 2 |
|----------|--------|---------|
| A. $a(1+2+3)$ | → | 5. $a + 2a + 3a$ |
| B. $2(12 - 4)$ | → | 2. $12 \cdot 2 - 4 \cdot 2$ |
| C. $12a + 3b$ | → | 1. $3(4a + b)$ |
| D. $\frac{2}{3}(15a - 18)$ | → | 6. $10a - 12$ |
| E. $6a + 10b$ | → | 3. $2(3a + 5b)$ |
| F. $0.4(5 - 2.5a)$ | → | 7. $2 - a$ |
| G. $2a + 3a$ | → | 4. $(2 + 3)a$ |

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✔ Answer:

- A → 5
- B → 2
- C → 1
- D → 6
- E → 3
- F → 7
- G → 4

Let me know if you'd like a diagram or visual explanation!