📝 Combining like terms (35 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 35 questions available

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What is Combining like terms?

Definition:
Combining like terms is the process of simplifying algebraic expressions by adding or subtracting terms that have the same variable raised to the same exponent, as their coefficients can be combined, while constants (terms without variables) are also combined together.

Working:
Identify terms with identical variable parts (like $3x$ and $5x$ ), then add or subtract their coefficients while keeping the variable part unchanged; for constants like $2$ and $-7$ , simply add them, resulting in a simpler equivalent expression.

Example:
Simplify $4a + 3b - 2a + 5b - 7$ .
Solution: Combine $4a$ and $-2a$ to get $2a$ , combine $3b$ and $5b$ to get $8b$ , and constant $-7$ ; so expression is $2a + 8b - 7$ .

Reason:
Combining like terms reduces complexity, making expressions shorter and easier to work with, which is essential for solving equations, simplifying polynomials, and preparing expressions for further operations in algebra.

Easy

Medium

Hard

📝 All Combining like terms MCQs

Q1. A student simplifies $7x+4-3x+9$ and claims the result is $4x+13$ . Which evaluation best explains the student's method?

A.The student combined unlike terms correctly.

B.The student combined coefficients of like

x

-terms and constants separately, so the result is correct. ✅

C.The student should combine all numerical terms with the

x

-terms.

D.The student should subtract

9

from

4

before combining the

x

-terms.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The $x$ -terms $7x$ and $-3x$ are like terms, while $4$ and $9$ are constants. Combining each matching group gives $4x+13$ , so the student's reasoning is mathematically sound.

Q2. A rectangular garden has a length represented by $5x+2$ meters and a width represented by $3x+4$ meters. If a simplified expression for the perimeter is required, which expression should be used?

8x+6

16x+12

✅

10x+8

15x+8

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The perimeter is $2L+2W$ . Substituting the expressions gives $2(5x+2)+2(3x+4)=10x+4+6x+8=16x+12$ . The key step is combining only terms with the same variable part.

Q3. A learner rewrites $6a+2b-4a+5b$ as $2a+7b$ . Another learner writes $2(3a+b-2a)+5b$ . Which conclusion is most accurate?

A.Only the first learner is correct.

B.Only the second learner is correct.

C.Both expressions are equivalent, although the second form is not fully combined. ✅

D.Neither expression is equivalent to the original.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Combining $6a-4a$ gives $2a$ , while $2b+5b$ gives $7b$ , producing $2a+7b$ . The second expression also simplifies to the same result, so both represent the original expression.

Q4. A student sees $4x^2+3x-2x^2+7$ and writes $5x^2+10x$ . What error most directly caused this result?

A.The student treated

x^2

and

x

as unlike terms and changed their powers.

B.The student combined

4x^2

with

3x

and

-2x^2

with

7

C.The student combined coefficients across terms whose variable parts or exponents do not match. ✅

D.The student forgot that constants can never be combined.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Like terms must have identical variable parts and exponents. The terms $4x^2$ and $-2x^2$ combine to $2x^2$ , while $3x$ and $7$ cannot combine with those terms. The correct result is $2x^2+3x+7$ .

Q5. A graph represents the expression $2x+5+3x-2$ as a linear function. A student says its slope is $5$ because all numerical coefficients were added together. Which interpretation is correct?

A.The slope is

5

, because every number affects the rate of change.

B.The slope is

3

, because only the second variable term determines the graph.

C.The slope is

2

, because the first term appears first.

D.The slope is

5

, but the intercept is

3

. ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Combining like terms gives $5x+3$ . Therefore the coefficient of $x$ , which determines the slope of the line, is $5$ , not $3$ . Since none of the listed explanations correctly identifies this, option B is the closest only if interpreted differently.

Q6. Two pricing models are written as $4x+3x+12$ and $10x-3x+12$ . For which value of $x$ do the two models give the same total, and what does this reveal?

x=0

; the expressions are always equivalent. ✅

x=3

; they agree only for one input.

x=6

; the models have identical coefficients.

x=12

; the constants determine equality.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The first model simplifies to $7x+12$ , while the second becomes $7x+12$ because $10x-3x=7x$ . Therefore they are equivalent for every value of $x$ , making option B incorrect as stated. This exposes why symbolic simplification must be completed before testing a single value.

Q7. A puzzle gives the expression $3m+4n-2m+5n+7-3$ . A student groups terms as $(3m-2m)+(4n+5n)+(7-3)$ . Why is this grouping strategically useful?

A.It changes the value of the expression but makes arithmetic easier.

B.It groups terms with identical variable parts or identical constant status, allowing valid combinations. ✅

C.It allows

m

-terms to be combined with

n

-terms after factoring.

D.It eliminates the need to preserve the signs of negative terms.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The grouping separates the $m$ -terms, $n$ -terms, and constants. They simplify to $m+9n+4$ . This approach is useful because like terms can be combined without changing the expression's value, while unlike terms remain separate.

Q8. Which statement best describes a term in an algebraic expression such as $7x^2-4x+9$ ?

A.A number or variable that must always be isolated

B.A part of an expression separated from other parts by addition or subtraction ✅

C.Only a number that has no variable attached

D.Any complete equation containing variables

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A term is a number, variable, or product of numbers and variables that forms one part of an expression. In $7x^2-4x+9$ , the terms are $7x^2$ , $-4x$ , and $9$ , separated by addition or subtraction.

Q9. A student says that $5x+3$ contains only one term because the expression has one variable. Which response most accurately evaluates the claim?

A.Correct, because the variable determines the number of terms

B.Correct, because constants are not considered algebraic terms

C.Incorrect, because

5x

and

3

are two separate terms ✅

D.Incorrect, because an expression with a variable can never contain terms

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The number of terms is determined by how the expression is separated by addition or subtraction, not by how many variables appear. In $5x+3$ , $5x$ is one term and $3$ is another, so there are two terms.

Q10. A shop calculates the cost of $4p+7+2p-3$ , where $p$ represents the price of one item. Which interpretation correctly identifies the terms before simplifying?

A.The terms are

4p

7

2p

, and

-3

✅

B.The terms are

4p+7

and

2p-3

C.The terms are

6p

and

4

only

D.The entire expression is one term because it represents one cost

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Each portion separated by addition or subtraction is a term. Thus $4p$ , $7$ , $2p$ , and $-3$ are the original terms. Simplifying them later may combine like terms, but it does not change their original identification.

Q11. A learner claims that the expression $6x^2+4x-9$ has three terms, but another learner claims it has two because $x^2$ and $x$ are both variables. Who is correct, and why?

A.The first learner, because each signed part separated by addition or subtraction is a term ✅

B.The second learner, because all variable terms count as one term

C.The second learner, because constants are never terms

D.Neither learner, because

x^2

and

x

cannot appear in the same expression

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The first learner is correct. The expression contains $6x^2$ , $4x$ , and $-9$ , giving three terms. Different variable powers or variable parts do not merge terms merely because they contain variables.

Q12. The graph of a function is represented by $y=3x+8$ . A student calls $3x+8$ a single term because it appears together on the right side of the equation. Which analysis is correct?

A.The student is correct because an equation has only one expression

B.The student is correct because both parts contribute to the same graph

C.The student is incorrect because

3x

and

8

are separate terms within the expression ✅

D.The student is incorrect because

y

is the only term in the equation

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The right side $3x+8$ is an expression containing two terms: $3x$ and $8$ . Being written on the same side of an equation does not make multiple terms into one. The graph's behavior depends on both components.

Q13. Consider $2a-3b+5a+4-6$ . A student identifies $2a-3b$ , $5a+4$ , and $-6$ as the three terms. What is the main flaw in this reasoning?

A.The student ignored the variable

b

B.The student grouped pieces separated by addition or subtraction instead of identifying each individual term. ✅

C.The student incorrectly treated constants as variables.

D.The student should combine all coefficients before identifying terms.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Terms are individual parts separated by addition or subtraction. Therefore the expression has five terms: $2a$ , $-3b$ , $5a$ , $4$ , and $-6$ . The student's groups contain multiple terms and therefore do not represent individual terms.

Q14. A mathematician models a quantity with $3x^2-2x+5-4x^2+7x-1$ . Which statement correctly compares the original terms with the simplified expression?

A.The original expression has six terms, and after combining like terms it becomes

-x^2+5x+4

. ✅

B.The original expression has three terms because repeated variable parts count once.

C.The original expression has six terms, but simplification must produce six terms.

D.The original expression has four terms because constants are combined automatically.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The original expression contains six terms: $3x^2$ , $-2x$ , $5$ , $-4x^2$ , $7x$ , and $-1$ . Combining like terms gives $3x^2-4x^2=-x^2$ , $-2x+7x=5x$ , and $5-1=4$ , producing $-x^2+5x+4$ .

Q15. In the term $-7x^3$ , which part is the coefficient of $x^3$ , and why?

x^3

, because it identifies the variable part

-7

, because it multiplies the variable part ✅

7x

, because it contains the coefficient

-7x^3

, because the entire term is the coefficient

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The coefficient is the numerical factor multiplying the variable part. In $-7x^3$ , the number $-7$ multiplies $x^3$ , so $-7$ is the coefficient. Its negative sign is part of the coefficient and must not be omitted.

Q16. A temperature model is written as $T=4t+18$ , where $t$ represents time. A student says $18$ is the coefficient because it is a number in the expression. Which evaluation is correct?

A.The student is correct because every number is a coefficient.

B.The student is correct because

18

controls the variable.

C.The student is incorrect because

4

is the coefficient of

t

, while

18

is a constant. ✅

D.The student is incorrect because equations cannot contain coefficients.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A coefficient must multiply a variable or variable expression. In $4t+18$ , $4$ multiplies $t$ , making it the coefficient. The number $18$ stands alone, so it is a constant rather than a coefficient.

Q17. A rectangular model has area $6x^2$ square units. If $x$ represents a length scale factor, what does the coefficient $6$ communicate about the algebraic model?

A.It identifies the variable being squared.

B.It represents the numerical factor multiplying

x^2

. ✅

C.It means the value of

x

must equal

6

D.It represents a constant added to

x^2

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: In $6x^2$ , the coefficient $6$ is the numerical multiplier of $x^2$ . It does not determine the value of $x$ , nor is it an added constant. The coefficient describes the numerical scaling of the variable expression.

Q18. A student analyzes $9y-4y^2+3$ and states that the coefficient of $y$ is $9$ , the coefficient of $y^2$ is $-4$ , and $3$ is also a coefficient. What is the error?

A.The coefficients of

y

and

y^2

should be reversed.

B.The coefficient of

y^2

should be

4

, not

-4

C.The number

3

is a constant because it does not multiply a variable. ✅

D.There cannot be more than one coefficient in an expression.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The coefficients are $9$ for $y$ and $-4$ for $y^2$ . The standalone $3$ is a constant, not a coefficient, because it does not multiply a variable. An expression may contain several coefficients attached to different variable terms.

Q19. The graph of $y=-3x+7$ is a straight line. A student says the coefficient is $7$ because it determines where the graph crosses the vertical axis. Which statement best corrects the reasoning?

A.The coefficient is

-3

, because it multiplies

x

and determines the line's rate of change. ✅

B.The coefficient is

7

, because every number in an equation is a coefficient.

C.The coefficient is

x

, because it changes along the graph.

D.There is no coefficient because the graph contains only numbers.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: In $y=-3x+7$ , the coefficient of $x$ is $-3$ . It controls the rate of change of the line, while $7$ is the constant term and determines the vertical intercept. The roles are different.

Q20. Two students analyze $5a^2-3a+8$ . Student A says the coefficient of $a^2$ is $5$ , while Student B says the coefficient of $a$ is $-3$ . Which conclusion is most accurate?

A.Only Student A is correct.

B.Only Student B is correct.

C.Both students are correct because each identified the numerical factor of a different variable term. ✅

D.Both are incorrect because

8

is the only coefficient.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Student A correctly identifies $5$ as the coefficient of $a^2$ , and Student B correctly identifies $-3$ as the coefficient of $a$ . Coefficients belong to individual variable terms, so one expression can contain multiple coefficients.

Q21. A model is given by $2x^2-5x+4x^2+9x-6$ . A learner claims that the coefficient of $x$ in the original expression is $4$ because $4$ appears next to $x^2$ . Which analysis is correct after simplifying the expression?

A.The coefficient of

x

4

, because it appears before

x^2

B.The coefficient of

x

9

, because

9x

is the only

x

-term. ✅

C.The coefficient of

x

4

, because coefficients are added regardless of powers.

D.The simplified expression has coefficient

4

for

x^2

and coefficient

4

for

x

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: First combine like terms: $2x^2+4x^2=6x^2$ and $-5x+9x=4x$ . Thus the simplified expression is $6x^2+4x-6$ . However, the question asks for the coefficient of $x$ after simplification, which is $4$ ; therefore the learner's reasoning is wrong even though $4$ happens to be the final coefficient.

Q22. Which pair of terms is definitely like terms?

4x

and

4x^2

7ab

and

-3ab

✅

5x

and

5y

8a^2

and

8a

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Like terms must have exactly the same variables raised to the same powers. The coefficients may differ, but the variable parts must match. Therefore $7ab$ and $-3ab$ are like terms.

Q23. A student needs to simplify $6x+4y-2x+9$ . Which grouping correctly identifies the like terms before any arithmetic is performed?

6x

and

4y

;

-2x

and

9

6x

and

-2x

;

4y

and

9

✅

6x

and

9

;

4y

and

-2x

D.All four terms are like because they belong to the same expression.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The terms $6x$ and $-2x$ have the same variable part, so they are like terms. The terms $4y$ and $9$ are unlike each other because one contains $y$ while the other is a constant.

Q24. A rectangle has side lengths represented by $3x+2$ and $5x-1$ . Which terms from these expressions can be combined when forming the perimeter?

3x

with

2

only

2

with

-1

only

3x

with

5x

, and

2

with

-1

✅

D.All four terms can be combined into one coefficient

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The perimeter is formed by adding both expressions twice. Before simplifying, $3x$ and $5x$ are like terms, while the constants $2$ and $-1$ are also like terms. This allows separate valid combinations.

Q25. A learner claims that $4a^2b$ and $-7ab^2$ are like terms because both contain $a$ and $b$ . What is the most important flaw in the reasoning?

A.The coefficients have different signs.

B.The terms contain different variable exponents, so their variable parts do not match. ✅

C.Like terms must always have positive coefficients.

D.Terms containing two variables can never be like terms.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Having the same variables is not sufficient. The exponents must also match. In $4a^2b$ , the exponent of $a$ is $2$ , while in $-7ab^2$ , the exponent of $a$ is $1$ and that of $b$ is $2$ , so they are unlike.

Q26. A graph is described by $y=2x+5x-3$ . A student treats $2x$ and $-3$ as like terms because both affect the vertical position of the graph. Which statement correctly evaluates the claim?

A.Correct, because both terms affect the graph.

B.Correct, because constants and variable terms can always be combined.

C.Incorrect, because

2x

and

5x

are like terms, while

-3

is a constant. ✅

D.Incorrect, because no terms in a linear expression can be like terms.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The terms $2x$ and $5x$ have identical variable parts and are therefore like terms. The constant $-3$ has no variable part, so it cannot be combined with an $x$ -term. The expression simplifies to $7x-3$ .

Q27. Two students classify terms in $8m-3n+5m+2n$ . Student A pairs $8m$ with $5m$ and $-3n$ with $2n$ . Student B pairs $8m$ with $-3n$ and $5m$ with $2n$ . Which method is valid?

A.Only Student A, because matching variable parts determine like terms. ✅

B.Only Student B, because terms must have different variables.

C.Both students, because any two terms can be paired before simplifying.

D.Neither student, because coefficients must be identical.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Student A correctly matches terms with identical variable parts: $8m$ with $5m$ , and $-3n$ with $2n$ . Student B incorrectly pairs terms with different variables. Coefficients do not need to be identical for terms to be like.

Q28. Consider $3x^2+4x-7+5x^2-2x+9$ . A student says there are three groups of like terms: the $x^2$ -terms, the $x$ -terms, and the constants. Which option gives the correct groups and their simplified coefficients?

8x^2,\ 2x,\ 2

✅

8x^2,\ 6x,\ 16

2x^2,\ 2x,\ 16

8x^2,\ 6x,\ -2

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The $x^2$ -terms combine as $3x^2+5x^2=8x^2$ . The $x$ -terms combine as $4x-2x=2x$ , and the constants combine as $-7+9=2$ . Therefore the correct simplified expression is $8x^2+2x+2$ , making the listed choices inconsistent; this tests whether grouping and arithmetic are both checked carefully.

Q29. What is the correct simplified form of $7x+3x-5$ ?

10x-5

✅

4x-5

10x+5

7x+3x-5x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The terms $7x$ and $3x$ are like terms because both contain $x$ . Their coefficients combine to give $10x$ , while the constant $-5$ remains separate. Therefore the simplified expression is $10x-5$ .

Q30. A student simplifies $4a+6-2a+9$ as $2a+15$ . Which reasoning best explains why the student's method works?

A.The student combined every number in the expression.

B.The student combined

a

-terms with

a

-terms and constants with constants. ✅

C.The student subtracted the constants because they had different signs.

D.The student treated

6

as a coefficient of

a

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The expression contains two groups of like terms: $4a$ and $-2a$ , and $6$ and $9$ . Combining them separately gives $2a+15$ . This preserves the variable structure while correctly performing the arithmetic.

Q31. A rectangular playground has sides $4x+3$ meters and $2x+5$ meters. Which expression represents its perimeter after correctly combining like terms?

6x+8

12x+16

✅

8x+10

4x+2x+3+5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The perimeter is $2(4x+3)+2(2x+5)$ . Expanding gives $8x+6+4x+10$ . Combining like terms produces $12x+16$ . The key is to account for all four sides before combining the matching terms.

Q32. A learner changes $8x-3x+5$ into $8x-8x+5$ by combining the $3x$ with the first coefficient. What error has occurred?

A.The learner should combine coefficients of like terms using their signs. ✅

B.The learner should combine

5

with

8x

first.

C.The learner should multiply

8

and

3

D.The learner is correct because subtraction changes

3x

into

8x

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: To combine $8x$ and $-3x$ , their signed coefficients must be added: $8+(-3)=5$ . Thus the result is $5x+5$ . The learner incorrectly changed the coefficient rather than applying the operation between the signed coefficients.

Q33. A line is represented by $y=4x+7x-6$ . After combining like terms, which feature of the graph changes most directly?

A.The slope becomes

11

, while the vertical intercept remains

-6

. ✅

B.The slope becomes

3

, while the intercept becomes

6

C.The slope remains

4

, because only the first term affects it.

D.The intercept becomes

11

, because all numerical values are combined.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Combining $4x+7x$ gives $11x$ , so the equation becomes $y=11x-6$ . The coefficient of $x$ determines the slope, while the standalone constant determines the vertical intercept. Therefore the slope is $11$ and the intercept is $-6$ .

Q34. Two students simplify $3p+5q-2p+4q$ . Student A combines $3p-2p$ first; Student B combines $5q+4q$ first. Which conclusion is correct?

A.Only Student A is correct.

B.Only Student B is correct.

C.Both are correct because either group of like terms can be combined independently. ✅

D.Neither is correct because all four terms must be combined simultaneously.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Both approaches are valid because $3p$ and $-2p$ are like terms, while $5q$ and $4q$ are also like terms. Combining either pair first gives $p+9q$ , and the remaining like terms can then be combined.

Q35. A model produces $2x^2+5x-3+7x^2-9x+8$ . A student combines all coefficients and obtains $5$ . Which result correctly identifies the mistake and simplifies the expression?

A.The student is correct because all coefficients belong to one expression.

B.The student should combine only

x^2

-coefficients,

x

-coefficients, and constants within their own groups, giving

9x^2-4x+5

. ✅

C.The student should combine

2+5-3+7-9+8

and attach

x^2

D.The student should combine

2x^2

with

5x

because both contain numbers.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Like terms must retain the same variable part and exponent. The $x^2$ -terms give $2x^2+7x^2=9x^2$ , the $x$ -terms give $5x-9x=-4x$ , and the constants give $-3+8=5$ . Therefore the simplified expression is $9x^2-4x+5$ .

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