Algebra Proofs Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve and complete the Algebra Proofs Notes worksheet step by step.

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✔ Part 1: Fill in the Blanks

We'll go through each definition and fill in the missing words.

#### • Proof – A logical explanation of why something is true.
> *A proof is a logical argument that shows a statement is true based on known facts.*

#### • Two-Column Proof – A type of proof with two columns: Statements and Reasons.
> *This format organizes the steps of the proof clearly: one column for what you're saying (statements), and one for why it’s valid (reasons).*

#### • Statements – Information you are saying is true.
> *Each line in the left column is a claim or step in the logic.*

#### • Reasons – Explanation of why the statement is true.
> *The right column justifies each statement using properties, definitions, or previously established facts.*

#### • Given – The information you know to be true.
> *This is the starting point of the proof—what is assumed or provided.*

#### • Prove – The information you are trying to show.
> *This is the conclusion you want to reach using logical steps.*

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✔ Part 2: Label the Example

Now, let’s label the example:

```
__________: 3x = 15
__________: x = 5
```

- The first line is Given: 3x = 15
- The second line is Prove: x = 5

So we write:
> Given: 3x = 15
> Prove: x = 5

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✔ Part 3: Complete the Two-Column Proof Table

We have this table:

| Statements | Reasons |
|----------------|-------------|
| 1. 3x = 15 | 1. |
| 2. $\frac{3x}{3} = \frac{15}{3}$ | 2. |
| 3. x = 5 | 3. |

Now, fill in the Reasons column:

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#### Step 1:
Statement: 3x = 15
Reason: Given
> This is the starting information.

#### Step 2:
Statement: $\frac{3x}{3} = \frac{15}{3}$
Reason: Division Property of Equality
> We divided both sides of the equation by 3. This property states that if $a = b$, then $\frac{a}{c} = \frac{b}{c}$ (as long as $c \neq 0$).

#### Step 3:
Statement: x = 5
Reason: Simplification / Division of Numbers
> Or more precisely: Simplify both sides (or Arithmetic)
> Since $\frac{15}{3} = 5$, and $\frac{3x}{3} = x$, so $x = 5$.

Alternatively, some teachers prefer:
> Reason: Simplify fractions or Division of whole numbers

But in formal proofs, often it's acceptable to say:
> Reason: Simplify

However, the most precise reason here is:
> Reason: Simplify both sides of the equation

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✔ Final Completed Worksheet

Here’s the full filled-in version:

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ALGEBRA PROOFS notes

- Proof – A logical explanation of why something is true.
- Two-Column Proof – A type of proof with two columns: Statements and Reasons.
- Statements – Information you are saying is true.
- Reasons – Explanation of why the statement is true.
- Given – The information you know to be true.
- Prove – The information you are trying to show.

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Example:

Given: 3x = 15
Prove: x = 5

| Statements | Reasons |
|----------------|-------------|
| 1. 3x = 15 | 1. Given |
| 2. $\frac{3x}{3} = \frac{15}{3}$ | 2. Division Property of Equality |
| 3. x = 5 | 3. Simplify (or Simplify both sides) |

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✔ Summary of Key Concepts

- A two-column proof uses Statements and Reasons.
- Start with Given information.
- Use properties of equality (like addition, subtraction, multiplication, division) to manipulate equations.
- End with the Prove statement.
- Each step must be justified with a valid reason.

This structure helps build logical reasoning skills essential in algebra and geometry.

Let me know if you'd like a printable version or more examples!