Question

Fill in the blanks!
What is the sum of the algebraic expressions -2.1x + 3.7 and 5 + 4.9x ?

1. Identify like terms.
2. Rewrite using properties of operations when necessary.
-2.1x + 3.7 + 5 + 4.9 x

Commutative property
-2.1x + _ + 5 + 3.7

Distributive property
x (_ + 4.9) + 5 + 3.7

3. Simplify
2.8x + _

4. Check by substitution.
x = 1
-2.1 (1) + 3.7
-2.1 + 3.7 = _
2.8 (1) + 8.7
= _
5 + 4.9 (1)
5 + 4.9 = _
9.9 + 1.6 = 11.5

Our correct expression, 2.8x + 8.7, is the sum of the two algebraic expressions.

Answer 1

Let's go through each step of your prompt.

1. **Identify like terms.**
The like terms in the expression \(-2.1x + 3.7 + 5 + 4.9x\) are:
- \( -2.1x\) and \( 4.9x\) (these are the terms with \(x\))
- \(3.7\) and \(5\) (these are the constant terms)

2. **Rewrite using properties of operations when necessary.**
We can rearrange it using the commutative property:
\[
-2.1x + 4.9x + 3.7 + 5
\]
Applying the distributive property is unnecessary here, but we can prepare it as follows:
\[
(-2.1 + 4.9)x + (3.7 + 5)
\]

3. **Simplify.**
Simplify the coefficients:
\[
(-2.1 + 4.9)x = 2.8x
\]
And for the constant terms:
\[
(3.7 + 5) = 8.7
\]
So now we combine them to give:
\[
2.8x + 8.7
\]

4. **Check by substitution.**
Let's evaluate both forms with \(x = 1\).

- For the original expression:
\[
-2.1(1) + 3.7 = -2.1 + 3.7 = 1.6
\]
\[
5 + 4.9(1) = 5 + 4.9 = 9.9
\]
Sum:
\[
1.6 + 9.9 = 11.5
\]

- For the simplified expression:
\[
2.8(1) + 8.7 = 2.8 + 8.7 = 11.5
\]

Since both calculations yield \(11.5\), the original expression and the simplified expression are equivalent.

Thus, the correct expression is indeed:
\[
\boxed{2.8x + 8.7}
\]

Answer 2

You didn't fill in the blanks

Answer 3

Apologies for not filling in the blanks clearly! Let’s go through your prompt again and fill in each of the provided blanks step-by-step.

**1. Identify like terms.**
The expression is:
\(-2.1x + 3.7 + 5 + 4.9x\)
Like terms are:
- \( -2.1x\) and \( 4.9x\) (terms with \(x\))
- \(3.7\) and \(5\) (constant terms)

**2. Rewrite using properties of operations when necessary.**
Original expression:
\(-2.1x + 3.7 + 5 + 4.9x\)

Using the **Commutative Property**:
Starting with the like terms,
\(-2.1x + 4.9x + 3.7 + 5\)

**Distributive Property:**
\(-2.1x + 4.9x\) can be grouped:
\[
x(-2.1 + 4.9) + (3.7 + 5)
\]

So, we can fill in the blank as:
\[
x( _ + 4.9) + 5 + 3.7
\]
The blank is \( -2.1 \).

**3. Simplify.**
Now, simplifying:
- Combine the coefficients:
\(-2.1 + 4.9 = 2.8\)
- Combine the constants:
\(3.7 + 5 = 8.7\)

Thus the final expression is:
\[
2.8x + 8.7
\]
This means you'll fill in the blank as:
\[
2.8x + _
\]
The blank is \( 8.7 \).

**4. Check by substitution.**
When \(x = 1\):
- From the original expression:
\[
-2.1(1) + 3.7 = -2.1 + 3.7 = 1.6
\]
\[
5 + 4.9(1) = 5 + 4.9 = 9.9
\]
Total:
\[
1.6 + 9.9 = 11.5
\]

- From the simplified expression:
\[
2.8(1) + 8.7 = 2.8 + 8.7 = 11.5
\]

Both calculations confirm that the sums are correct.

So, here’s the partially filled prompt:

1. Identify like terms.
- Blank (like terms for constants): **3.7 + 5**

2. Rewrite using properties of operations when necessary.
Using the **Commutative property**:
\(-2.1x + **4.9** + 5 + 3.7\)
Using the **Distributive property**:
\(x(**-2.1** + 4.9) + 5 + 3.7\)

3. Simplify:
- \(2.8x + **8.7**\)

I hope this format meets your expectations better! Thank you for your patience!

Answer 4

You still haven't filled in these blanks:

4. Check by substitution.
x = 1
-2.1 (1) + 3.7
-2.1 + 3.7 = _
2.8 (1) + 8.7
= _
5 + 4.9 (1)
5 + 4.9 = _
9.9 + 1.6 = 11.5

Answer 5

Thank you for your clarification! Let’s fill in the blanks step by step for the substitution check.

**4. Check by substitution.**

For \( x = 1 \):

1. Calculate the original expression:
\[
-2.1(1) + 3.7
\]
\[
-2.1 + 3.7 = 1.6 \quad \text{(This fills in the first blank)}
\]

2. Calculate the simplified expression:
\[
2.8(1) + 8.7
\]
\[
2.8 + 8.7 = 11.5 \quad \text{(This fills in the second blank)}
\]

3. Calculate the second part of the original expression:
\[
5 + 4.9(1)
\]
\[
5 + 4.9 = 9.9 \quad \text{(This fills in the third blank)}
\]

4. Add the results from the first part and the second part of the original expression:
\[
9.9 + 1.6 = 11.5
\]

So here are the filled-in blanks:

**Check by substitution:**
- \( -2.1(1) + 3.7 \)
\( -2.1 + 3.7 = **1.6** \)

- \( 2.8(1) + 8.7 \)
\( = **11.5** \)

- \( 5 + 4.9(1) \)
\( 5 + 4.9 = **9.9** \)

Thank you again for your patience! I hope this completes it as intended!