You’ve uploaded a worksheet with three separate algebraic proofs, each requiring you to fill in the “Reasons” column based on the “Statements.” Let’s solve each one step by step and explain the reasoning.
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Problem 1: Given: 2x - 10 = 20; Prove: x = 15
Statements:
1. 2x - 10 = 20
2. 2x - 10 + 10 = 20 + 10
3. 2x = 30
4. (2x)/2 = 30/2
5. x = 15
✔ Reasons:
1.
Given — This is the starting equation provided in the problem.
2.
Addition Property of Equality — You add 10 to both sides to isolate the term with x.
3.
Simplification (or Arithmetic) — Combining like terms: -10 + 10 cancels out, 20 + 10 = 30.
4.
Division Property of Equality — Divide both sides by 2 to solve for x.
5.
Simplification (or Arithmetic) — 30 ÷ 2 = 15.
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Problem 2: Given: x/4 + 5 = 12; Prove: x = 28
Statements:
1. x/4 + 5 = 12
2. x/4 + 5 - 5 = 12 - 5
3. x/4 = 7
4. x/4 × 4 = 7 × 4
5. x = 28
✔ Reasons:
1.
Given — Starting equation.
2.
Subtraction Property of Equality — Subtract 5 from both sides to isolate the fraction.
3.
Simplification — 5 - 5 = 0, 12 - 5 = 7.
4.
Multiplication Property of Equality — Multiply both sides by 4 to eliminate the denominator.
5.
Simplification — 7 × 4 = 28.
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Problem 3: Given: 4(x + 3) = 16; Prove: x = 1
Statements:
1. 4(x + 3) = 16
2. 4x + 12 = 16
3. 4x + 12 - 12 = 16 - 12
4. 4x = 4
5. x = 1
✔ Reasons:
1.
Given — Starting equation.
2.
Distributive Property — Distribute the 4: 4 × x = 4x, 4 × 3 = 12.
3.
Subtraction Property of Equality — Subtract 12 from both sides to isolate the term with x.
4.
Simplification — 12 - 12 = 0, 16 - 12 = 4 → so 4x = 4.
5.
Division Property of Equality — Divide both sides by 4 → x = 1.
*(Note: The worksheet says "x = 1" as the final answer, which is correct. If your worksheet has a typo or misnumbered step, this is still the correct logic.)*
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✔ Final Answer Summary:
Problem 1 Reasons:
1. Given
2. Addition Property of Equality
3. Simplification
4. Division Property of Equality
5. Simplification
Problem 2 Reasons:
1. Given
2. Subtraction Property of Equality
3. Simplification
4. Multiplication Property of Equality
5. Simplification
Problem 3 Reasons:
1. Given
2. Distributive Property
3. Subtraction Property of Equality
4. Simplification
5. Division Property of Equality
Let me know if you’d like these formatted into a table or need help writing them into your worksheet!
Parent Tip: Review the logic above to help your child master the concept of geometry algebraic proofs worksheet.
