Let's solve each algebraic proof step by step and fill in the
"Reasons" column with appropriate justifications for each statement. These proofs follow the structure of
two-column proofs, where each
statement is justified by a
reason (such as "Given", "Addition Property of Equality", etc.).
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Problem 1:
Given: $2x - 10 = 20$
Prove: $x = 15$
| Statements | Reasons |
|-----------|--------|
| 1. $2x - 10 = 20$ | 1.
Given |
| 2. $2x - 10 + 10 = 20 + 10$ | 2.
Addition Property of Equality (add 10 to both sides) |
| 3. $2x = 30$ | 3.
Simplification (combine like terms) |
| 4. $\frac{2x}{2} = \frac{30}{2}$ | 4.
Division Property of Equality (divide both sides by 2) |
| 5. $x = 15$ | 5.
Simplification |
✔ Explanation: We start with the given equation, add 10 to both sides to isolate the term with $x$, then divide both sides by 2 to solve for $x$. Each step uses a property of equality.
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Problem 2:
Given: $\frac{x}{2} + 5 = 12$
Prove: $x = 14$
| Statements | Reasons |
|-----------|--------|
| 1. $\frac{x}{2} + 5 = 12$ | 1.
Given |
| 2. $\frac{x}{2} + 5 - 5 = 12 - 5$ | 2.
Subtraction Property of Equality (subtract 5 from both sides) |
| 3. $\frac{x}{2} = 7$ | 3.
Simplification |
| 4. $\frac{x}{2} \cdot 2 = 7 \cdot 2$ | 4.
Multiplication Property of Equality (multiply both sides by 2) |
| 5. $x = 14$ | 5.
Simplification |
✔ Explanation: We subtract 5 from both sides to isolate the fraction, then multiply both sides by 2 to eliminate the denominator and solve for $x$.
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Problem 3:
Given: $4(x + 3) = 16$
Prove: $x = 1$
| Statements | Reasons |
|-----------|--------|
| 1. $4(x + 3) = 16$ | 1.
Given |
| 2. $4x + 12 = 16$ | 2.
Distributive Property (distribute 4 to $x$ and 3) |
| 3. $4x + 12 - 12 = 16 - 12$ | 3.
Subtraction Property of Equality (subtract 12 from both sides) |
| 4. $4x = 4$ | 4.
Simplification |
| 5. $\frac{4x}{4} = \frac{4}{4}$ | 5.
Division Property of Equality (divide both sides by 4) |
| 6. $x = 1$ | 6.
Simplification |
✔ Explanation: First, we use the distributive property to expand the left side. Then, subtract 12 from both sides to isolate the variable term, and finally divide both sides by 4 to solve for $x$.
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✔ Summary of Key Properties Used:
-
Given: The starting information.
-
Addition/Subtraction Property of Equality: You can add or subtract the same value from both sides of an equation.
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Multiplication/Division Property of Equality: You can multiply or divide both sides by the same non-zero number.
-
Distributive Property: $a(b + c) = ab + ac$
-
Simplification: Combining like terms or reducing fractions.
These are standard algebraic reasoning steps used in formal proofs.
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Parent Tip: Review the logic above to help your child master the concept of geometry algebraic proofs worksheet.
