Two-column Algebraic Proofs Practice Worksheet for solving equations and justifying each step with mathematical reasoning.

Let’s solve each equation step by step and fill in the two-column proof with correct statements and reasons.

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Problem 1: -4x + 10 = -5x + 18 → Prove: x = 8

We start with the given equation.

a. -4x + 10 = -5x + 18 → Given
b. Add 5x to both sides → to get all x terms on one side
→ -4x + 5x + 10 = -5x + 5x + 18 → x + 10 = 18
Reason: Addition Property of Equality
c. Subtract 10 from both sides → to isolate x
→ x + 10 - 10 = 18 - 10 → x = 8
Reason: Subtraction Property of Equality
d. x = 8 → Substitution Prop. (already given)

✔ So for Problem 1:
b. x + 10 = 18 → Addition Property of Equality
c. x = 8 → Subtraction Property of Equality

Wait — but step d is already “x = 8” with reason “Substitution Prop.” That seems odd. Usually, substitution isn’t needed here. Maybe it’s a typo or mislabel? But we’ll follow the worksheet.

Actually, looking again — step d says “x = 8” and reason “Substitution Prop.” — that doesn’t make sense unless they’re substituting something. Perhaps it’s meant to be “Simplification” or just “Final Answer”. But since the worksheet says “Substitution Prop.” for d, maybe they consider replacing “x + 10 = 18” with “x = 8” as substitution? Unlikely. Probably a mistake in the worksheet.

But let’s stick to standard algebra:

After step c: x = 8 — that’s the solution. So perhaps step d is redundant? Or maybe they want us to write the final answer there?

Looking at the structure:

They have:
a. Given
b. ?
c. ?
d. x = 8 → Substitution Prop.

That suggests step d is the conclusion. So maybe steps b and c lead to x=8, and then d is restating it? That’s unusual.

Alternatively, perhaps step d is meant to be the final line after simplifying.

Let me re-solve carefully:

Start: -4x + 10 = -5x + 18
Add 5x to both sides: (-4x + 5x) + 10 = (-5x + 5x) + 18 → x + 10 = 18 → Reason: Addition Property of Equality
Subtract 10: x + 10 - 10 = 18 - 10 → x = 8 → Reason: Subtraction Property of Equality

So step d should be x=8 with reason Subtraction Property — but the worksheet says “Substitution Prop.” for d. That must be an error. Since we’re filling blanks, we’ll put the correct reason for step c, and leave d as is? No — we need to fill b and c.

Perhaps the worksheet intends:

b. x + 10 = 18 → Addition Property of Equality
c. x = 8 → Subtraction Property of Equality
d. x = 8 → Substitution Prop. ← this is redundant, but maybe they want it?

I think it’s a mistake. For educational purposes, I’ll correct it logically.

But since the user wants the worksheet filled, I’ll provide what makes mathematical sense.

For Problem 1:

b. x + 10 = 18 → Addition Property of Equality
c. x = 8 → Subtraction Property of Equality
(d is already filled)

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Problem 2: 4x = 12x + 32 → Prove: x = -4

a. 4x = 12x + 32 → Given
b. Subtract 12x from both sides → 4x - 12x = 12x - 12x + 32 → -8x = 32 → Reason: Subtraction Property of Equality
c. Divide both sides by -8 → -8x / -8 = 32 / -8 → x = -4 → Reason: Division Property of Equality
d. x = -4 → Substitution Prop. (again, probably should be Division Prop., but worksheet says Substitution for d? Wait, no — in problem 2, d is blank? Let me check.

Looking back at image:

Problem 2 has:
a. Given
b. _____
c. _____
d. _____ ← wait, no — in the image, problem 2 only has a, b, c — no d? Let me recount.

In the original image description:

Problem 2:
Statements: a, b, c — only three lines? But problem 1 has a,b,c,d.

Actually, looking at the text:

"2) 4x = 12x + 32 Prove: x = -4
Statements Reasons
a. 4x = 12x + 32 Given
b. ________________________________
c. ________________________________"

Only up to c? But then it says "d." in problem 1. Inconsistency.

In the user's text:

For problem 2:
"a. 4x = 12x + 32 Given
b. ________________________________
c. ________________________________"

No d. So probably only two steps after given.

Similarly, problem 3 has a,b,c,d — four lines.

Problem 4 has a,b,c,d,e — five lines.

So we fill accordingly.

For Problem 2:

a. Given
b. -8x = 32 → Subtraction Property of Equality (subtract 12x)
c. x = -4 → Division Property of Equality (divide by -8)

Yes.

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Problem 3: (3x+5)/2 = 7 → Prove: x=3

a. (3x+5)/2 = 7 → Given
b. Multiply both sides by 2 → 3x+5 = 14 → Reason: Multiplication Property of Equality
c. Subtract 5 from both sides → 3x = 9 → Reason: Subtraction Property of Equality
d. Divide both sides by 3 → x = 3 → Reason: Division Property of Equality

Perfect.

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Problem 4: 5(x+2) = -3x -6 → Prove: x=-2

Note: In the worksheet, it says "5(x+2) = -3x + 6" — but that must be a typo, because if it's +6, then solving:

5x + 10 = -3x + 6 → 8x = -4 → x = -0.5, not -2.

But the prove is x=-2, so likely it should be -6, not +6.

Check the image description: "4) 5(x + 2) = -3x - 6 Prove: x = -2" — yes, in the initial problem statement it says "-6", but in the table it says "+6". Typo in the table.

Looking back at user input:

"4) 5(x + 2) = -3x - 6 Prove: x = -2" — in the text above the table.

But in the table: "a. 5(x + 2) = -3x + 6 Given" — inconsistency.

This is critical. If it's +6, then:

5(x+2) = -3x +6
5x +10 = -3x +6
8x = -4
x = -0.5 ≠ -2

But prove is x=-2, so it must be -6.

Probably a typo in the worksheet. We'll assume it's -6 as stated in the problem title.

So:

a. 5(x+2) = -3x -6 → Given
b. Distribute: 5x + 10 = -3x -6 → Reason: Distributive Property
c. Add 3x to both sides: 5x + 3x + 10 = -6 → 8x + 10 = -6 → Reason: Addition Property of Equality
d. Subtract 10: 8x = -16 → Reason: Subtraction Property of Equality
e. Divide by 8: x = -2 → Reason: Division Property of Equality

Yes, matches prove.

If it were +6, it wouldn't work. So we go with -6.

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Problem 5: 4 + 2(3x + 5) = 11 - x → Prove: x = -3/7

a. Given
b. Distribute: 4 + 6x + 10 = 11 - x → Reason: Distributive Property
c. Combine like terms: 6x + 14 = 11 - x → Reason: Simplification (or Combine Like Terms)
d. Add x to both sides: 7x + 14 = 11 → Reason: Addition Property of Equality
e. Subtract 14: 7x = -3 → Reason: Subtraction Property of Equality
f. Divide by 7: x = -3/7 → Reason: Division Property of Equality

The worksheet has up to e, so probably:

a. Given
b. 4 + 6x + 10 = 11 - x → Distributive Property
c. 6x + 14 = 11 - x → Combine Like Terms
d. 7x + 14 = 11 → Addition Property of Equality
e. 7x = -3 → Subtraction Property of Equality
Then x = -3/7 would be next, but not listed? The prove is x=-3/7, so perhaps e is 7x=-3, and we stop, implying division.

But the worksheet has "e." as last line, so likely e is 7x = -3, and the prove is understood.

To match, we'll fill:

b. 4 + 6x + 10 = 11 - x → Distributive Property
c. 6x + 14 = 11 - x → Combine Like Terms
d. 7x + 14 = 11 → Addition Property of Equality
e. 7x = -3 → Subtraction Property of Equality

And x = -3/7 is the result, though not explicitly written, but the prove is given.

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Problem 6: 3 + 4(x - 2) = 27 → Prove: x=8

a. Given
b. Distribute: 3 + 4x - 8 = 27 → Reason: Distributive Property
c. Combine like terms: 4x - 5 = 27 → Reason: Simplification
d. Add 5: 4x = 32 → Reason: Addition Property of Equality
e. Divide by 4: x = 8 → Reason: Division Property of Equality

Worksheet has up to d? Let's see:

"6) 3 + 4(x - 2) = 27 Prove: x = 8
Statements Reasons
a. 3 + 4(x - 2) = 27 Given
b. ________________________________
c. ________________________________
d. ________________________________"

Only up to d? But we need five steps including given.

a is given, so b,c,d are three more.

But to get to x=8, we need:

After distribute and combine: 4x -5 =27
Then 4x=32
Then x=8

So perhaps:

b. 3 + 4x - 8 = 27 → Distributive Property
c. 4x - 5 = 27 → Combine Like Terms
d. 4x = 32 → Addition Property of Equality
Then x=8 is implied, but not listed. The prove is x=8, so maybe d is 4x=32, and we assume next step.

But the worksheet has "d." as last, so likely d is 4x=32, and the prove is the conclusion.

To be precise, we'll fill what's needed.

Since prove is x=8, and d is the last statement, perhaps d should be x=8, but that skips steps.

Better to have:

b. 4x - 5 = 27 → after distributing and combining
c. 4x = 32 → add 5
d. x = 8 → divide by 4

Yes, that fits.

So:

b. 4x - 5 = 27 → Distributive Property and Combine Like Terms (or two steps, but we can combine)

Actually, distributing gives 3 + 4x - 8 = 27, then combining 3-8=-5, so 4x -5 =27.

So reason for b could be "Distributive Property and Combine Like Terms" or split.

But to save space, often combined.

We'll do:

b. 4x - 5 = 27 → Distributive Property and Simplification
c. 4x = 32 → Addition Property of Equality
d. x = 8 → Division Property of Equality

Perfect.

Now, compiling all answers.

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Final Answers for each problem:

Problem 1:
b. x + 10 = 18 → Addition Property of Equality
c. x = 8 → Subtraction Property of Equality
(Note: d is already filled as x=8 with Substitution Prop., which is likely incorrect, but we don't change given.)

Problem 2:
b. -8x = 32 → Subtraction Property of Equality
c. x = -4 → Division Property of Equality

Problem 3:
b. 3x + 5 = 14 → Multiplication Property of Equality
c. 3x = 9 → Subtraction Property of Equality
d. x = 3 → Division Property of Equality

Problem 4: (assuming typo, should be -6 not +6)
b. 5x + 10 = -3x - 6 → Distributive Property
c. 8x + 10 = -6 → Addition Property of Equality
d. 8x = -16 → Subtraction Property of Equality
e. x = -2 → Division Property of Equality

Problem 5:
b. 4 + 6x + 10 = 11 - x → Distributive Property
c. 6x + 14 = 11 - x → Combine Like Terms
d. 7x + 14 = 11 → Addition Property of Equality
e. 7x = -3 → Subtraction Property of Equality

Problem 6:
b. 4x - 5 = 27 → Distributive Property and Simplification
c. 4x = 32 → Addition Property of Equality
d. x = 8 → Division Property of Equality

Now, for the final answer, since the user didn't specify which problem, but the task is to solve each, I'll provide the completed worksheets.

But the instruction says "provide the final correct answer" — probably meaning the filled-in parts.

Since it's multiple problems, I'll list them clearly.

Final Answer:

For Problem 1:
b. Statement: x + 10 = 18, Reason: Addition Property of Equality
c. Statement: x = 8, Reason: Subtraction Property of Equality

For Problem 2:
b. Statement: -8x = 32, Reason: Subtraction Property of Equality
c. Statement: x = -4, Reason: Division Property of Equality

For Problem 3:
b. Statement: 3x + 5 = 14, Reason: Multiplication Property of Equality
c. Statement: 3x = 9, Reason: Subtraction Property of Equality
d. Statement: x = 3, Reason: Division Property of Equality

For Problem 4:
b. Statement: 5x + 10 = -3x - 6, Reason: Distributive Property
c. Statement: 8x + 10 = -6, Reason: Addition Property of Equality
d. Statement: 8x = -16, Reason: Subtraction Property of Equality
e. Statement: x = -2, Reason: Division Property of Equality

For Problem 5:
b. Statement: 4 + 6x + 10 = 11 - x, Reason: Distributive Property
c. Statement: 6x + 14 = 11 - x, Reason: Combine Like Terms
d. Statement: 7x + 14 = 11, Reason: Addition Property of Equality
e. Statement: 7x = -3, Reason: Subtraction Property of Equality

For Problem 6:
b. Statement: 4x - 5 = 27, Reason: Distributive Property and Simplification
c. Statement: 4x = 32, Reason: Addition Property of Equality
d. Statement: x = 8, Reason: Division Property of Equality

Note: In Problem 4, the worksheet shows "+6" but it should be "-6" to prove x=-2. We used "-6" as per the problem statement.