Properties of Equality worksheet - Free Printable

Let’s go through each table one by one and fill in the missing steps or equations using the correct property of equality.

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First Table:

Equation | Steps
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x + (x - 6) = 8 | Original Equation
2x - 6 = 8 | Distributive Property? Wait — actually, this is combining like terms. But note: x + (x - 6) becomes 2x - 6. That’s just simplifying by combining x + x. So it’s not distributive — it’s Combining Like Terms. But since the worksheet says “Properties of Equality”, we need to check if that counts. Actually, combining like terms is an algebraic simplification, not a property of equality. However, sometimes worksheets accept “Simplification” or “Combine Like Terms”. But let’s see what comes next.

Next step:
They say “Addition Property of Equality” leads to x = 7.

So from 2x - 6 = 8, to get to x = 7, they must have added 6 to both sides first:

2x - 6 + 6 = 8 + 6 → 2x = 14

Then divide by 2: x = 7

But the table skips 2x = 14 and goes straight to x = 7 after saying “Addition Property of Equality”.

Wait — that doesn’t make sense. Addition Property would give you 2x = 14, then Division Property gives x = 7.

Looking at the table again:

Row 1: x + (x - 6) = 8 → Original
Row 2: 2x - 6 = 8 → ?
Row 3: [blank] → Addition Property of Equality
Row 4: x = 7 → ?

Actually, Row 3 should be 2x = 14 (after adding 6 to both sides), and Row 4 is dividing by 2.

But the table has only 4 rows total. Let me re-read:

It says:

Equation | Steps
x + (x - 6) = 8 | Original Equation
2x - 6 = 8 | ???
[blank equation] | Addition Property of Equality
x = 7 | ???

So the blank equation must be 2x = 14 (because you add 6 to both sides of 2x - 6 = 8).

Then from 2x = 14 to x = 7, that’s Division Property of Equality.

But the last row doesn’t have a step labeled — so maybe we’re supposed to fill in the missing equation AND the missing step?

Wait — looking back at the user’s image description, the tables have some cells blank. We need to fill in the blanks.

In first table:

- Second row step is missing → going from x+(x-6)=8 to 2x-6=8 → that’s Combining Like Terms (but technically not a property of equality). Some curricula call this “Simplify” or “Algebraic Simplification”. Since the worksheet says “Properties of Equality”, perhaps they expect us to skip labeling that? Or maybe it’s accepted.

Actually, let’s look at other tables for pattern.

Second table:

x + (4x + 12) = 12 → Original
5x + 12 = 22 → ???
5x = 10 → ???
Division Property → x = 2? Wait no, last line isn't shown but implied.

Wait in second table:

After 5x + 12 = 22, next is 5x = 10 → that’s Subtraction Property (subtract 12 from both sides)

Then 5x = 10 to x = 2 would be Division Property — but in the table it says “Division Property of Equality” is the step for going to the next equation — which must be x = 2.

But in the given table, the last equation isn’t written — wait no, in the user’s text, for second table:

"5x = 10" then next row is empty equation with step "Division Property of Equality" — so the equation should be x = 2.

Similarly, in first table, after “Addition Property of Equality”, the equation is blank — so we need to write 2x = 14.

Then from 2x = 14 to x = 7, the step is Division Property of Equality — but that cell is blank in the table? In the user’s text, for first table, last row is “x = 7” with no step listed — so we might need to fill that too.

Actually, reviewing the original problem statement: “Identify the property of equality that justifies each missing step or equation”

So sometimes the equation is missing, sometimes the step is missing.

Let’s tackle each table systematically.

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Table 1:

Given:

1. x + (x - 6) = 8 → Original Equation
2. 2x - 6 = 8 → Step missing
3. [equation missing] → Addition Property of Equality
4. x = 7 → Step missing

Step 2: From x + (x - 6) to 2x - 6 — this is combining like terms. While not strictly a “property of equality”, many textbooks list this as “Simplify” or allow it under algebraic manipulation. Since the worksheet focuses on properties of equality, and this step doesn’t involve changing both sides equally, perhaps it’s acceptable to label it as “Combine Like Terms” or leave it? But the instruction says “property of equality”, so maybe they don’t require labeling this one? Looking at other tables, similar steps are present without being labeled as properties — e.g., in table 2, going from x+(4x+12)=12 to 5x+12=22 — that’s also combining like terms.

In table 2, the step for that is not provided — only later steps are labeled.

Perhaps for steps that are simplifications (combining like terms, distributing), we don’t label them with equality properties, and only label when we perform an operation on both sides.

But in table 1, row 2 step is blank — so we must fill it.

I think for consistency, since combining like terms is not a property of equality, but the worksheet might still expect us to write something. Alternatively, maybe they consider it part of the process.

Looking ahead — in table 4, they have:

5x + (3x + 4) = 28 → Original
5x + 3x + 4 = 28 → step missing — that’s distributive? No, removing parentheses — associative/commutative? Actually, it’s just rewriting — probably “Remove Parentheses” or “Simplify”.

Then 8x + 4 = 28? Wait no, next is 8x + 12 = 28 — that doesn’t match.

Wait in table 4:

5x + (3x + 4) = 28
→ 5x + 3x + 4 = 28 [step missing]
→ 8x + 12 = 28 [step missing?] — wait, 5x+3x is 8x, but 4 should stay 4, not become 12. That seems wrong.

Hold on — there might be a typo in my reading.

User wrote for table 4:

"5x + (3x + 4) = 28"
"5x + 3x + 4 = 28"
"8x + 12 = 28" — that can’t be right because 4 ≠ 12.

Unless... oh! Perhaps it's 5x + 3(x + 4) = 28? But user wrote "(3x + 4)".

Let me double-check the user’s input:

For fourth table:

"5x + (3x + 4) = 28"
"5x + 3x + 4 = 28"
"8x + 12 = 28" — this must be a mistake. 5x + 3x is 8x, and +4 should be +4, not +12.

Unless the original was 5x + 3(x + 4) = 28, then distributing: 5x + 3x + 12 = 28, then 8x + 12 = 28.

That makes sense. Probably a typo in the user’s transcription — it should be 3(x + 4), not (3x + 4).

Because otherwise 8x + 12 = 28 doesn't follow from 5x + 3x + 4 = 28.

So I’ll assume it’s 5x + 3(x + 4) = 28.

Similarly, in table 1, everything else checks out.

Back to table 1.

To resolve, let’s fill based on standard expectations.

Table 1:

- Row 2: 2x - 6 = 8 — this comes from simplifying left side: x + x - 6 = 2x - 6. The justification is Combine Like Terms (though not a property of equality, it’s often accepted in such contexts).

- Row 3: After Addition Property of Equality — we add 6 to both sides of 2x - 6 = 8 → 2x = 14. So equation is 2x = 14

- Row 4: From 2x = 14 to x = 7 — divide both sides by 2 → Division Property of Equality

So fill:

Row 2 step: Combine Like Terms (or Simplify)
Row 3 equation: 2x = 14
Row 4 step: Division Property of Equality

But since the worksheet says “Properties of Equality”, and Combine Like Terms isn’t one, perhaps they leave it unlabeled? But the cell is blank, so we must fill.

Looking at table 2:

x + (4x + 12) = 12 → Original
5x + 12 = 22 → step missing — again, combining like terms: x + 4x = 5x, so 5x + 12 = 22? But original right side is 12, not 22. That doesn’t match.

User wrote: "x + (4x + 12) = 12" then "5x + 12 = 22" — that can’t be, because if you combine left side, you get 5x + 12 = 12, not 22.

There’s inconsistency.

Perhaps it’s x + 4(x + 3) = 12 or something.

Let’s calculate: if x + (4x + 12) = 12, then 5x + 12 = 12, then 5x = 0, x=0.

But next is 5x + 12 = 22 — which suggests the original equation might be different.

Another possibility: maybe it’s x + 4x + 12 = 22 originally? But user said "=12".

This is confusing.

Perhaps in table 2, the original is x + (4x + 12) = 22? Then 5x + 12 = 22, then subtract 12: 5x = 10, then divide: x=2.

And the step "Division Property of Equality" leads to x=2.

In the user’s text, for table 2, after "5x = 10", it says "Division Property of Equality" and then presumably x=2, but not written.

Similarly, in table 1, after "Addition Property", we have blank equation, which should be 2x=14, then to x=7 with Division Property.

For table 2, the step from x+(4x+12)=12 to 5x+12=22 is problematic unless the original is =22.

I think there might be typos in the user’s transcription, but based on common problems, I’ll assume:

For table 2: original is x + (4x + 12) = 22, then 5x + 12 = 22 (combine like terms), then 5x = 10 (Subtraction Property), then x = 2 (Division Property).

But user wrote "=12" for original, then "=22" for next — likely a mistake; probably original is =22.

To proceed, I'll use the numbers as given and find logical steps.

Perhaps in table 2, the "22" is a typo and should be "12", but then 5x+12=12, 5x=0, etc.

But then "5x = 10" wouldn't follow.

Another idea: maybe it's x + 4(x + 3) = 12 or something.

Let's stop overcomplicating. I'll solve based on the most logical interpretation for each table, assuming minor typos are corrected for coherence.

Final decision for each table:

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Table 1:

- Step for 2x - 6 = 8: Combine Like Terms (even though not a property of equality, it's necessary)
- Equation after Addition Property: 2x = 14 (added 6 to both sides)
- Step for x = 7: Division Property of Equality (divided by 2)

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Table 2:

Assume original is x + (4x + 12) = 22 (since next is 5x+12=22)

- Step for 5x + 12 = 22: Combine Like Terms
- Step for 5x = 10: Subtraction Property of Equality (subtracted 12 from both sides)
- Equation after Division Property: x = 2

But in user's text, for table 2, the last equation isn't specified, but the step is given as "Division Property of Equality", so the equation must be x=2.

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Table 3:

4(x - 6) = 40 → Original
x - 6 = 10 → ?
x = 16 → ?

From 4(x-6)=40 to x-6=10: divided both sides by 4 → Division Property of Equality

From x-6=10 to x=16: added 6 to both sides → Addition Property of Equality

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Table 4:

Assume it's 5x + 3(x + 4) = 28 (to make 8x+12=28 make sense)

- 5x + 3(x + 4) = 28 → Original
- 5x + 3x + 12 = 28 → Distributive Property
- 8x + 12 = 28 → Combine Like Terms
- Then next: they have "Subtraction Property of Equality" leading to ? , then x=12

From 8x + 12 = 28, subtract 12: 8x = 16 → that's Subtraction Property

Then divide by 8: x = 2, but user says x=12 — contradiction.

If x=12, then from 8x + 12 = 28, 8*12=96, 96+12=108≠28.

Mistake.

Perhaps it's 5x + 3x + 4 = 28, then 8x + 4 = 28, then 8x = 24, x=3.

But user has "8x + 12 = 28" and "x=12".

Another possibility: the equation is 5x + 3(x + 4) = 108 or something.

Let's calculate what would give x=12.

Suppose after subtraction, we have 8x = 96, then x=12.

So before subtraction, 8x + 12 = 108? But user says =28.

This is inconsistent.

Perhaps "8x + 12 = 28" is a typo, and it's "8x + 12 = 108" or " something else.

Maybe it's 5x + 3x + 4 = 28, then 8x + 4 = 28, then 8x = 24, x=3, but user says x=12.

I think there's a typo in the user's input for table 4.

Common problem: 5x + 3(x + 4) = 28 → 5x + 3x + 12 = 28 → 8x + 12 = 28 → 8x = 16 → x=2.

But user has x=12.

Another common one: 5x + 3(x + 4) = 108 → 8x + 12 = 108 → 8x = 96 → x=12.

Yes! Likely the original is =108, not =28.

Because 8*12 + 12 = 96+12=108.

So probably "5x + (3x + 4) = 108" but user wrote 28 by mistake.

Or "5x + 3(x + 4) = 108".

I'll assume that for coherence.

So for table 4:

- 5x + 3(x + 4) = 108 → Original
- 5x + 3x + 12 = 108 → Distributive Property
- 8x + 12 = 108 → Combine Like Terms
- 8x = 96 → Subtraction Property of Equality (subtract 12)
- x = 12 → Division Property of Equality

But in user's text, it's written as "=28" for the first three lines, then "x=12", so likely typo.

To match x=12, I'll use 108.

Since the user might have copied wrong, but for solving, I'll use the numbers that work.

Perhaps in the actual image, it's 108, but user typed 28.

I think for accuracy, I'll solve with the numbers that make sense mathematically.

So for table 4, assume the constant is such that x=12 works.

From x=12, and last step is after Subtraction Property, so before that, 8x = 96, so before subtraction, 8x + 12 = 108.

So original equation should be 5x + 3(x + 4) = 108.

Then:

- 5x + 3(x + 4) = 108 → Original
- 5x + 3x + 12 = 108 → Distributive Property
- 8x + 12 = 108 → Combine Like Terms
- 8x = 96 → Subtraction Property of Equality
- x = 12 → Division Property of Equality

Perfect.

Now, back to filling all blanks.

Let's compile the answers for each table.

Table 1:
- Missing step for "2x - 6 = 8": Combine Like Terms
- Missing equation for "Addition Property of Equality": 2x = 14
- Missing step for "x = 7": Division Property of Equality

Table 2:
Assume original is x + (4x + 12) = 22 (so that 5x+12=22 makes sense)
- Missing step for "5x + 12 = 22": Combine Like Terms
- Missing step for "5x = 10": Subtraction Property of Equality
- Missing equation for "Division Property of Equality": x = 2

But in user's text, for table 2, the original is written as "=12", but then next is "=22", so perhaps it's a different setup.

Another possibility: maybe it's x + 4x + 12 = 22 originally, but user grouped it as x + (4x + 12) = 12 by mistake.

To avoid confusion, I'll use the numbers as given in the sequence.

In table 2:
- First equation: x + (4x + 12) = 12
- Second: 5x + 12 = 22 — this is impossible unless there's a calculation error.

Perhaps the "22" is "12", and "5x = 10" is "5x = 0", but then division gives x=0.

But user has "5x = 10", so likely the original right side is 22.

I think it's safe to assume that in table 2, the original equation is x + (4x + 12) = 22, and "12" is a typo.

Similarly for table 4, "28" is a typo for "108".

For the sake of providing a correct answer, I'll proceed with the mathematically consistent versions.

So final answers per table:

Table 1:
- Step for 2x - 6 = 8: Combine Like Terms
- Equation for Addition Property: 2x = 14
- Step for x = 7: Division Property of Equality

Table 2:
- Step for 5x + 12 = 22: Combine Like Terms
- Step for 5x = 10: Subtraction Property of Equality
- Equation for Division Property: x = 2

Table 3:
- Step for x - 6 = 10: Division Property of Equality
- Step for x = 16: Addition Property of Equality

Table 4:
- Step for 5x + 3x + 12 = 108: Distributive Property (assuming 3(x+4))
- Step for 8x + 12 = 108: Combine Like Terms
- Equation for Subtraction Property: 8x = 96
- Step for x = 12: Division Property of Equality

But in user's text for table 4, the equations are given as:
"5x + (3x + 4) = 28" — which should be 5x + 3x + 4 = 28, then 8x + 4 = 28, then 8x = 24, x=3, not 12.

To match x=12, it must be different.

Perhaps it's 5x + 3(x + 4) = 108, as I said.

I think for the response, I'll provide the answers based on standard problems, and note if there are discrepancies.

Since the user asked to solve accurately, I'll use the numbers that work.

So for table 4, I'll assume the constant is 108.

Now, to write the final answer as per the format.

The Final Answer should be the filled-in values for the blanks.

Since the question is to identify the property or equation for each missing part, I'll list them table by table.

But the response format requires a single Final Answer section.

Perhaps I can summarize the missing parts.

Since it's a worksheet, the expected output is to fill the blanks.

For clarity, I'll state the missing items.

Final Answer:

For Table 1:
- The step for "2x - 6 = 8" is "Combine Like Terms".
- The equation for "Addition Property of Equality" is "2x = 14".
- The step for "x = 7" is "Division Property of Equality".

For Table 2:
- The step for "5x + 12 = 22" is "Combine Like Terms".
- The step for "5x = 10" is "Subtraction Property of Equality".
- The equation for "Division Property of Equality" is "x = 2".

For Table 3:
- The step for "x - 6 = 10" is "Division Property of Equality".
- The step for "x = 16" is "Addition Property of Equality".

For Table 4:
- The step for "5x + 3x + 12 = 108" is "Distributive Property".
- The step for "8x + 12 = 108" is "Combine Like Terms".
- The equation for "Subtraction Property of Equality" is "8x = 96".
- The step for "x = 12" is "Division Property of Equality".

Note: In Table 4, the original equation is assumed to be 5x + 3(x + 4) = 108 to make the solution x=12 correct, as the given "=28" appears to be a typo.

But to match the user's input exactly, if we must use "=28", then x=12 is incorrect, but since the worksheet has x=12, I'll go with the corrected version.

For the purpose of this response, I'll box the key missing parts as per the tables.

Since the Final Answer should be concise, I'll list the missing equations and steps in order.

Perhaps:

The missing parts are:
- Table 1, row 2 step: Combine Like Terms
- Table 1, row 3 equation: 2x = 14
- Table 1, row 4 step: Division Property of Equality
- Table 2, row 2 step: Combine Like Terms
- Table 2, row 3 step: Subtraction Property of Equality
- Table 2, row 4 equation: x = 2
- Table 3, row 2 step: Division Property of Equality
- Table 3, row 3 step: Addition Property of Equality
- Table 4, row 2 step: Distributive Property
- Table 4, row 3 step: Combine Like Terms
- Table 4, row 4 equation: 8x = 96
- Table 4, row 5 step: Division Property of Equality

But this is lengthy.

Since the user may have specific blanks, and to keep it simple, I'll provide the answers as per the most reasonable interpretation.

For the Final Answer section, I'll write the filled values for the blanks in the order they appear.

But the instruction is to provide only the final correct answer after reasoning.

Perhaps for this type of problem, the Final Answer is the completed tables, but that's not practical.

I think the best way is to state the missing items clearly.

Given the constraints, I'll box the essential corrections.

After careful thought, here is the accurate solution based on standard mathematical principles and correcting obvious typos for coherence:

Final Answer:

Table 1:
- Missing step (row 2): Combine Like Terms
- Missing equation (row 3): 2x = 14
- Missing step (row 4): Division Property of Equality

Table 2:
- Missing step (row 2): Combine Like Terms
- Missing step (row 3): Subtraction Property of Equality
- Missing equation (row 4): x = 2

Table 3:
- Missing step (row 2): Division Property of Equality
- Missing step (row 3): Addition Property of Equality

Table 4:
- Missing step (row 2): Distributive Property
- Missing step (row 3): Combine Like Terms
- Missing equation (row 4): 8x = 96
- Missing step (row 5): Division Property of Equality

(Note: For Table 4, the constants are adjusted to 108 instead of 28 to ensure x=12 is correct, as 28 would yield x=2.)