Algebraic expressions worksheet for students to practice identifying variables, constants, and coefficients.

Let’s solve this step by step.

We are given algebraic expressions and asked to identify:

1. Variables – letters that stand for unknown numbers (like x, y, z).
2. Constants – plain numbers that don’t change (like 5, -3, 0.7).
3. Coefficients – the number in front of a variable (in “4x”, 4 is the coefficient).

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Part 1: Write the variables and constants

#### Expression 1:
x + ½ y – z + 4

- Variables: x, y, z → these are letters representing unknowns.
- Constants: 4 → it’s just a number with no letter.
→ Note: ½ is part of the term with y, so it’s not a constant — it’s a coefficient!

✔ So:
- Variables: x, y, z
- Constants: 4

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#### Expression 2:
a² – b³ + 2m

- Variables: a, b, m → all letters.
- Constants: none? Wait — there’s no standalone number. The “2” is attached to m, so it’s a coefficient.
→ So no pure constants here.

✔ So:
- Variables: a, b, m
- Constants: (none) → we can write “none” or leave blank, but since the worksheet has lines, probably expect “none”

But let’s check the format — maybe they want us to list what exists. If nothing, say “none”.

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#### Expression 3:
0.5q – 6p – 9qr – 1

Break down each term:
- 0.5q → variable q, coefficient 0.5
- -6p → variable p, coefficient -6
- -9qr → variables q and r, coefficient -9
- -1 → constant

So:
- Variables: q, p, r (note: qr means both q and r are variables)
- Constants: -1

Wait — should we list each variable once? Yes, even if repeated.

So variables: p, q, r
Constants: -1

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#### Expression 4:
½h – 7

Terms:
- ½h → variable h, coefficient ½
- -7 → constant

So:
- Variables: h
- Constants: -7

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Part 2: Write the coefficient of each variable

#### Expression 1:
3mn – 8n² + 7n² – 5

First, combine like terms if possible? But the question says “coefficient of each variable” — meaning per term as written? Or after simplifying?

Looking at the expression:
It has two n² terms: -8n² and +7n² → together that’s -1n²

But the problem might want coefficients *as written*, before combining. Let’s see the instruction: “Write the coefficient of each variable.”

Actually, looking at how the blanks are set up:

> Coefficient of n: ______
> Coefficient of n²: ______
> Coefficient of mn²: ______ ← wait, there’s no mn² term! That must be a typo? Or maybe it's testing attention?

Wait — original expression: 3mn – 8n² + 7n² – 5

There is NO “mn²” term. So coefficient of mn² is 0? Or perhaps it’s a mistake?

But let’s read carefully: the worksheet says:

> 1) 3mn – 8n² + 7n² – 5
> Coefficient of n: ___
> Coefficient of n²: ___
> Coefficient of mn²: ___ ← this seems wrong. Maybe it’s supposed to be “coefficient of mn”? Because “mn” is a term.

Possibly a typo in the worksheet. But since we have to answer based on what’s written...

Let’s assume it’s asking for:

- Coefficient of n → but “n” alone doesn’t appear. We have “mn” and “n²”. So unless they mean the total coefficient when combined... Hmm.

This is tricky. Let me think differently.

Perhaps they want the coefficient for each *type* of variable term present.

In 3mn – 8n² + 7n² – 5:

Term 1: 3mn → involves variables m and n → coefficient is 3
Term 2: -8n² → variable n squared → coefficient -8
Term 3: +7n² → another n² term → coefficient +7
Term 4: -5 → constant

Now, if we’re to report:

- Coefficient of n → but “n” by itself isn't a term. However, sometimes people refer to the power. Maybe they mean:

Actually, looking at the next problems, they ask for specific variables like “y”, “z”, etc., which are single variables.

Here, the terms are “mn”, “n²”, so perhaps:

They want:

- Coefficient of n → but again, not directly. Unless they consider “n” in “mn” — but that would be ambiguous.

Wait — maybe the third line is a typo and should be “coefficient of mn”?

Because otherwise, “mn²” doesn’t exist.

Given that, and to make sense, I’ll assume:

The intended questions are:

- Coefficient of mn → 3
- Coefficient of n² → (-8 + 7) = -1 (if combining) OR separately? But the blank is singular.

Looking at other problems, they seem to expect simplified or per-term.

But in problem 2 below, they have multiple terms with same variable.

Let’s look ahead.

Problem 2: 2b + 15c
Coefficient of b: 2
Coefficient of c: 15
That’s straightforward.

Problem 3: 10yz²y – 2 → wait, that’s messy. Probably 10y z² y? Which is 10 y² z²? Typo?

Actually, likely it’s “10y z²” or something. But let’s stick to first one.

To avoid confusion, let’s interpret as:

For expression 1: 3mn – 8n² + 7n² – 5

If we combine like terms: 3mn + (-8+7)n² -5 = 3mn - n² -5

Then:

- Coefficient of mn: 3
- Coefficient of n²: -1
- Coefficient of mn²: 0 (since no such term)

But the worksheet says “coefficient of mn²” — which doesn’t exist, so 0.

Alternatively, maybe it’s a misprint and should be “coefficient of m” or “n”.

But since we must answer as per given, I’ll go with:

After combining:

Expression becomes: 3mn - n² - 5

So:

- Coefficient of n → still not clear. Perhaps they mean for each distinct variable power combination.

I think safest is to treat each term as is, and assign coefficients to the variable parts mentioned.

But the blanks are labeled:

> Coefficient of n: ___
> Coefficient of n²: ___
> Coefficient of mn²: ___

Since “n” alone doesn’t appear, coefficient of n is 0? But that feels odd.

Another approach: in some contexts, “coefficient of n” means the multiplier when n is to the first power, regardless of other variables.

In “3mn”, n is to the first power, multiplied by 3m — but 3m is not a constant, so it’s not a numerical coefficient.

Ah — important point: coefficient usually refers to the numerical factor multiplying the variable(s).

So in “3mn”, the coefficient is 3, and the variable part is mn.

Similarly, in “-8n²”, coefficient is -8, variable part n².

So for the question “coefficient of n”, if they mean the term where n appears to the first power, that would be in “3mn”, but then the coefficient is 3, but it’s associated with mn, not just n.

This is ambiguous.

Looking at standard curriculum, when they say “coefficient of x” in an expression like 3xy + 2x, they might mean the total coefficient of terms containing x, but that’s advanced.

For middle school level, likely they want the numerical coefficient for each distinct variable term as written.

But here, the labels are confusing.

Let me check online or recall: typically in such worksheets, for an expression like 3mn - 8n² + 7n² - 5, they would ask:

- What is the coefficient of mn? → 3
- What is the coefficient of n²? → -8 and +7, but since there are two, perhaps they want them listed or combined.

But the blank is singular.

Perhaps for n², they want the net coefficient after combining: -8 + 7 = -1

And for "n", since no term has only n, coefficient is 0.

And for "mn²", no such term, so 0.

That makes sense.

So:

- Coefficient of n: 0 (no term with just n)
- Coefficient of n²: -1 (from -8 + 7)
- Coefficient of mn²: 0 (no such term)

But is "n" considered to include cases where n is with other variables? Usually not; "coefficient of n" means the multiplier when the term is proportional to n^1 and no other variables, or sometimes including other variables, but then it's not a constant coefficient.

I think for this level, they expect:

In 3mn, the coefficient is 3 for the term mn.

But the question is phrased as "coefficient of n", which is poorly worded.

Perhaps it's a typo, and it should be "coefficient of mn".

Given that, and to match common practice, I'll assume:

The intended questions are for the variable groups present.

But let's look at problem 3: "10yz²y – 2" — that's likely "10y z^2 y" which is 10 y^2 z^2, so coefficient of y^2 is 10, etc.

Perhaps for consistency, in problem 1, they want:

- For the term with n (first power): but there is no standalone n, so 0
- For n²: the combined coefficient -1
- For mn²: 0

I think that's the best we can do.

So for expression 1:

- Coefficient of n: 0
- Coefficient of n²: -1
- Coefficient of mn²: 0

But let's double-check with another approach.

Suppose we don't combine: then for n², there are two coefficients: -8 and +7, but the blank is one, so likely combine.

Similarly, in many textbooks, they simplify first.

So I'll go with combined form: 3mn - n² - 5

Then:

- The term with n to the first power: none, so coefficient of n is 0
- Term with n²: -n², so coefficient -1
- Term with mn²: none, so 0

Yes.

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#### Expression 2:
2b + 15c

Simple:

- Coefficient of b: 2
- Coefficient of c: 15

No issues.

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#### Expression 3:
10yz²y – 2

First, simplify: yz²y = y * z² * y = y² z²

So expression is: 10 y² z² - 2

Now, the questions are:

> Coefficient of y²: ___
> Coefficient of z²: ___
> Coefficient of y²z²: ___

Wait, the worksheet says:

> 3) 10yz²y – 2
> Coefficient of y²: ___
> Coefficient of z²: ___
> Coefficient of y²z²: ___

After simplifying, it's 10 y² z² - 2

So:

- Coefficient of y²: but y² alone isn't a term; it's y² z². So if they mean the coefficient when y² is considered, but it's multiplied by z², so technically, the coefficient of y² in the context of the whole term is 10 z², but that's not numerical.

Again, ambiguity.

Standard interpretation: the numerical coefficient of the term containing y² is 10, but only if we consider the entire variable part.

Typically, "coefficient of y²" would mean the multiplier of y² when other variables are treated as constants, but here it's mixed.

In this case, since the term is 10 y² z², the coefficient of y² z² is 10.

But they ask separately for y² and z².

This is problematic.

Perhaps they want:

- In the term 10 y² z², the coefficient for y² is 10 z², but that's not a number.

I think for this level, they might expect:

Since the term is 10 y² z², then:

- Coefficient of y²: 10 (assuming z² is part of the variable, but that's not accurate)

Better to think: the expression has only one variable term: 10 y² z²

So:

- If asked for coefficient of y², it's not defined alone; similarly for z².

But perhaps in the context, they mean the numerical coefficient associated with those powers.

Notice that in the answer blanks, they have "coefficient of y²", "coefficient of z²", "coefficient of y²z²"

For "coefficient of y²z²", it's clearly 10.

For "coefficient of y²", if we consider the term as having y², and z² is another variable, then the coefficient is 10 z², but again, not numerical.

I recall that in some curricula, for a term like 10 y² z², they say the coefficient of y² is 10 z², but that's for higher levels.

For middle school, likely they want the numerical coefficient for the entire term when specified.

Perhaps "coefficient of y²" means the number multiplying y² in the expression, but since y² is always with z², it's 10 z², which is not constant.

This is confusing.

Another idea: perhaps "10yz²y" is meant to be "10y z^2" and then "y" is separate, but that doesn't make sense.

Or maybe it's "10y * z^2 * y" = 10 y^2 z^2, as I had.

I think the safest is to assume that for "coefficient of y²", they mean in the context of the term, but since it's combined, perhaps they expect 10 for y²z², and for y² alone, it's not applicable.

But let's look at the pattern.

In problem 4: 4v - 5w + 3w - 8v

Which simplifies to (4v - 8v) + (-5w + 3w) = -4v -2w

Then coefficient of v: -4, w: -2, vw: 0

Similarly, for problem 3, after simplifying to 10 y² z² - 2, then:

- Coefficient of y²: if they mean the coefficient when the term is y², but it's not, so 0? But that seems wrong.

Perhaps "coefficient of y²" means the numerical factor for any term containing y², which is 10, and similarly for z², 10, and for y²z², 10.

That could be it.

In many introductory contexts, they might say the coefficient of y² is 10, implying in the term where it appears.

So for 10 y² z², the coefficient of y² is 10 (even though it's with z²), and similarly for z².

And for y²z², also 10.

That might be what they want.

To confirm, in problem 1, for 3mn, if they asked for coefficient of m, it would be 3, even though it's with n.

So yes, likely.

So for expression 3: 10 y² z² - 2

- Coefficient of y²: 10 (because in the term 10 y² z², y² has coefficient 10, with z² being part of the variable group, but they still attribute the number to y²)

Similarly, coefficient of z²: 10

Coefficient of y²z²: 10

And the -2 is constant, not affecting.

So:

- Coefficient of y²: 10
- Coefficient of z²: 10
- Coefficient of y²z²: 10

Yes, that makes sense for this level.

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#### Expression 4:
4v – 5w + 3w – 8v

Combine like terms:

v terms: 4v - 8v = -4v
w terms: -5w + 3w = -2w

So expression: -4v - 2w

Now:

- Coefficient of v: -4
- Coefficient of w: -2
- Coefficient of vw: 0 (no such term)

Perfect.

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Now, let's compile all answers.

First part: variables and constants

1) x + ½ y – z + 4
Variables: x, y, z
Constants: 4

2) a² – b³ + 2m
Variables: a, b, m
Constants: none → we can write "none" or leave, but since blank, perhaps "none"

3) 0.5q – 6p – 9qr – 1
Variables: p, q, r (since qr has both)
Constants: -1

4) ½h – 7
Variables: h
Constants: -7

Second part: coefficients

1) 3mn – 8n² + 7n² – 5 → simplified to 3mn - n² - 5
Coefficient of n: 0 (no term with just n)
Coefficient of n²: -1
Coefficient of mn²: 0 (no such term)

But earlier I thought maybe "coefficient of n" is misstated, but we'll go with this.

2) 2b + 15c
Coefficient of b: 2
Coefficient of c: 15

3) 10yz²y – 2 → 10 y² z² - 2
Coefficient of y²: 10
Coefficient of z²: 10
Coefficient of y²z²: 10

4) 4v – 5w + 3w – 8v → -4v -2w
Coefficient of v: -4
Coefficient of w: -2
Coefficient of vw: 0

Now, for the constants in part 1, for expression 2, should we write "none" or leave blank? Since the worksheet has lines, probably write "none".

Similarly, for coefficients, if 0, write 0.

Also, in expression 1 of part 2, "coefficient of n" — is it really 0? Let me see if there's another interpretation.

In some contexts, "coefficient of n" might mean the sum of coefficients of all terms containing n, but that would be for n in 3mn and in n², but n² is different power.

Usually, coefficient is per term type.

I think 0 is correct for "coefficient of n" since no term is k*n for constant k.

Similarly for others.

So final answers:

Part 1:

1) Variables: x, y, z; Constants: 4
2) Variables: a, b, m; Constants: none
3) Variables: p, q, r; Constants: -1
4) Variables: h; Constants: -7

Part 2:

1) Coeff of n: 0; Coeff of n²: -1; Coeff of mn²: 0
2) Coeff of b: 2; Coeff of c: 15
3) Coeff of y²: 10; Coeff of z²: 10; Coeff of y²z²: 10
4) Coeff of v: -4; Coeff of w: -2; Coeff of vw: 0

Now, to box the final answer as per instruction.

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

Final Answer:

Part 1:
1) Variables: x, y, z; Constants: 4
2) Variables: a, b, m; Constants: none
3) Variables: p, q, r; Constants: -1
4) Variables: h; Constants: -7

Part 2:
1) Coefficient of n: 0; Coefficient of n²: -1; Coefficient of mn²: 0
2) Coefficient of b: 2; Coefficient of c: 15
3) Coefficient of y²: 10; Coefficient of z²: 10; Coefficient of y²z²: 10
4) Coefficient of v: -4; Coefficient of w: -2; Coefficient of vw: 0