Worksheets · Algebra 1

Algebra 1 algebraic proofs worksheet

An algebraic proof solves an equation one step at a time and names the reason for each step: the Given, the Distributive Property, and the Addition, Subtraction, Multiplication and Division Properties of Equality. It is the warm-up for geometry proofs.

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Name

Algebraic Proofs

Date Period

Complete each proof.

  1. Solve and give a reason for each step: 4x+9=−27
  2. Solve and give a reason for each step: 5(x−9)=−25
  3. Solve and give a reason for each step: 2x−3=−1
  4. Solve and give a reason for each step: 2(x−7)=−12
  5. Solve and give a reason for each step: 2(x−6)=−22
  6. Solve and give a reason for each step: x3−5=−6
  7. Solve and give a reason for each step: x5+7=6
  8. Solve and give a reason for each step: 2x+6=−2
  9. Solve and give a reason for each step: 2(x+8)=8
  10. Solve and give a reason for each step: 5(x−3)=45
  11. Solve and give a reason for each step: 4(x+7)=68
  12. Solve and give a reason for each step: 6(x−9)=−24
  13. Solve and give a reason for each step: 2(x−3)=−8
  14. Solve and give a reason for each step: 4x−8=0
  15. Solve and give a reason for each step: 5(x+3)=−15
  16. Solve and give a reason for each step: 2(x−7)=−26
  17. Solve and give a reason for each step: 4x+2=−26
  18. Solve and give a reason for each step: 2(x+3)=−4
  19. Solve and give a reason for each step: x2+8=13
  20. Solve and give a reason for each step: 3x+3=9

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Algebraic ProofsAnswer keyVersion 1

Date Period

Complete each proof.

  1. Solve and give a reason for each step: 4x+9=−274x+9=−27 (Given); 4x=−36 (Subtraction Property of Equality); x=−9 (Division Property of Equality)
  2. Solve and give a reason for each step: 5(x−9)=−255(x−9)=−25 (Given); 5x−45=−25 (Distributive Property); 5x=20 (Addition Property of Equality); x=4 (Division Property of Equality)
  3. Solve and give a reason for each step: 2x−3=−12x−3=−1 (Given); 2x=2 (Addition Property of Equality); x=1 (Division Property of Equality)
  4. Solve and give a reason for each step: 2(x−7)=−122(x−7)=−12 (Given); 2x−14=−12 (Distributive Property); 2x=2 (Addition Property of Equality); x=1 (Division Property of Equality)
  5. Solve and give a reason for each step: 2(x−6)=−222(x−6)=−22 (Given); 2x−12=−22 (Distributive Property); 2x=−10 (Addition Property of Equality); x=−5 (Division Property of Equality)
  6. Solve and give a reason for each step: x3−5=−6x3−5=−6 (Given); x3=−1 (Addition Property of Equality); x=−3 (Multiplication Property of Equality)
  7. Solve and give a reason for each step: x5+7=6x5+7=6 (Given); x5=−1 (Subtraction Property of Equality); x=−5 (Multiplication Property of Equality)
  8. Solve and give a reason for each step: 2x+6=−22x+6=−2 (Given); 2x=−8 (Subtraction Property of Equality); x=−4 (Division Property of Equality)
  9. Solve and give a reason for each step: 2(x+8)=82(x+8)=8 (Given); 2x+16=8 (Distributive Property); 2x=−8 (Subtraction Property of Equality); x=−4 (Division Property of Equality)
  10. Solve and give a reason for each step: 5(x−3)=455(x−3)=45 (Given); 5x−15=45 (Distributive Property); 5x=60 (Addition Property of Equality); x=12 (Division Property of Equality)
  11. Solve and give a reason for each step: 4(x+7)=684(x+7)=68 (Given); 4x+28=68 (Distributive Property); 4x=40 (Subtraction Property of Equality); x=10 (Division Property of Equality)
  12. Solve and give a reason for each step: 6(x−9)=−246(x−9)=−24 (Given); 6x−54=−24 (Distributive Property); 6x=30 (Addition Property of Equality); x=5 (Division Property of Equality)
  13. Solve and give a reason for each step: 2(x−3)=−82(x−3)=−8 (Given); 2x−6=−8 (Distributive Property); 2x=−2 (Addition Property of Equality); x=−1 (Division Property of Equality)
  14. Solve and give a reason for each step: 4x−8=04x−8=0 (Given); 4x=8 (Addition Property of Equality); x=2 (Division Property of Equality)
  15. Solve and give a reason for each step: 5(x+3)=−155(x+3)=−15 (Given); 5x+15=−15 (Distributive Property); 5x=−30 (Subtraction Property of Equality); x=−6 (Division Property of Equality)
  16. Solve and give a reason for each step: 2(x−7)=−262(x−7)=−26 (Given); 2x−14=−26 (Distributive Property); 2x=−12 (Addition Property of Equality); x=−6 (Division Property of Equality)
  17. Solve and give a reason for each step: 4x+2=−264x+2=−26 (Given); 4x=−28 (Subtraction Property of Equality); x=−7 (Division Property of Equality)
  18. Solve and give a reason for each step: 2(x+3)=−42(x+3)=−4 (Given); 2x+6=−4 (Distributive Property); 2x=−10 (Subtraction Property of Equality); x=−5 (Division Property of Equality)
  19. Solve and give a reason for each step: x2+8=13x2+8=13 (Given); x2=5 (Subtraction Property of Equality); x=10 (Multiplication Property of Equality)
  20. Solve and give a reason for each step: 3x+3=93x+3=9 (Given); 3x=6 (Subtraction Property of Equality); x=2 (Division Property of Equality)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

3x+5=20

  1. Given.
  2. Subtract 5: 3x = 15, by the Subtraction Property of Equality.
  3. Divide by 3: x = 5, by the Division Property of Equality.

The answer is x=5.

Where students go wrong

Naming the operation instead of the property: "subtract 5" is the step, and the Subtraction Property of Equality is the reason.

Also called algebraic proofs, properties of equality or two column proof algebra.

Questions about these worksheets

Yes. Every algebraic proofs sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 1 is usually taken around 8th or 9th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the algebraic proofs questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Naming the operation instead of the property: "subtract 5" is the step, and the Subtraction Property of Equality is the reason.