📝 How to evaluate algebraic expressions (21 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 21 questions available

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What is How to evaluate algebraic expressions?

Definition:
Evaluating an algebraic expression means finding its numerical value by replacing each variable with its given number and then performing the operations according to the order of operations (PEMDAS), simplifying the expression step by step.

Working:
Substitute the given values for all variables in the expression, then apply PEMDAS: calculate inside parentheses, handle exponents, then multiplication/division from left to right, and finally addition/subtraction from left to right, simplifying to a single number.

Example:
Evaluate $2x^2 - 3y$ when $x = 3$ and $y = 4$ .
Solution: Substitute $x=3, y=4$ : $2(3)^2 - 3(4) = 2(9) - 12 = 18 - 12 = 6$ .

Reason:
This skill is fundamental in algebra because it connects abstract symbols to concrete numbers, allowing us to check solutions, graph functions, and apply formulas in physics, economics, and other fields.

Easy

Medium

Hard

📝 All How to evaluate algebraic expressions MCQs

Q1. If $x=4$ and $y=-2$ , what is the value of $3x^2-2xy+y^2$ ?

A.48

B.52

C.60

D.68 ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: First substitute $x=4$ and $y=-2$ into the entire expression. This gives $3(4^2)-2(4)(-2)+(-2)^2=48+16+4=68$ . Therefore, the correct value is $68$ , not $60$ .

Q2. A student evaluates $2a^2-3a+5$ for $a=-2$ and obtains $7$ . Which value should be obtained, and what mistake most likely caused the student's answer?

A.7; treating

a^2

-a^2

B.19; correctly applying the negative value before squaring ✅

C.15; adding the terms without preserving signs

D.11; replacing

a

with

2

instead of

-2

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Substituting $a=-2$ gives $2(-2)^2-3(-2)+5=8+6+5=19$ . The key error is often failing to recognize that the negative sign is included in the substitution but disappears when the value is squared.

Q3. A rectangular garden has length $x+3$ meters and width $x-1$ meters. If $x=6$ , a worker calculates the area as $6+3\times6-1$ . What is the correct area?

A.45 square meters ✅

B.54 square meters

C.63 square meters

D.72 square meters

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The dimensions must be evaluated separately before multiplying. When $x=6$ , the length is $9$ meters and the width is $5$ meters. Thus the area is $9\times5=45$ square meters, so option A is correct.

Q4. For $p=3$ , two methods are used for $4(p+2)-p^2$ . Method 1 gives $4(5)-9=11$ . Method 2 gives $20-3^2=17$ . Which conclusion is correct?

A.Method 1 is correct because addition comes before subtraction. ✅

B.Method 2 is correct because the final

p

should not be squared.

C.Both methods are correct because they use equivalent substitutions.

D.Neither method is correct because

4(p+2)

must be squared.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Substituting $p=3$ gives $4(3+2)-3^2=4(5)-9=20-9=11$ . Method 2 incorrectly changes the expression by treating $20-3^2$ as $20-9$ , which actually equals $11$ , not $17$ .

Q5. A graph shows the values of $f(x)=x^2-2x+3$ at $x=0,1,2,3$ as $3,2,3,6$ , respectively. Which statement correctly compares the values?

A.The expression has its greatest listed value at

x=1

B.The expression has equal values at

x=0

and

x=2

. ✅

C.The expression decreases continuously from

x=0

x=3

D.The value at

x=3

is smaller than the value at

x=0

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Evaluating the expression at the listed inputs produces $3,2,3,6$ . Therefore, the values at $x=0$ and $x=2$ are both $3$ . The graph would show the expression decreasing first and then increasing.

Q6. The expression $5m-2(n+3)+m^2$ is used to model a score. If $m=-1$ and $n=4$ , what score does the model produce?

A.-18

B.-16 ✅

C.-14

D.-12

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Substitute both values carefully: $5(-1)-2(4+3)+(-1)^2=-5-14+1=-18$ . The negative value of $m$ affects both $5m$ and $m^2$ , while the parentheses require $4+3$ to be evaluated first.

Q7. For $x=2$ , consider $E=3x^2-4(x-1)+2$ . A student says $E=14$ because $3x^2=12$ , $-4(x-1)=-4$ , and $2$ is added. Which evaluation correctly identifies the error?

A.The student is correct because

12-4+2=10

B.The error is calculating

3x^2

incorrectly; the correct value is

16

C.The error is evaluating

x-1

incorrectly; the correct value is

1

, giving

12-4+2=10

. ✅

D.The error is ignoring the coefficient

3

, giving

6-4+2=4

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: When $x=2$ , $3x^2=3(4)=12$ , while $x-1=1$ , so $-4(x-1)=-4$ . Therefore $E=12-4+2=10$ . The student's arithmetic conclusion of $14$ is incorrect even though the intermediate substitutions are mostly appropriate.

Q8. Which statement best describes what it means to evaluate an algebraic expression when values are assigned to its variables?

A.Rewrite every variable as a different variable

B.Replace variables with given values and simplify to obtain a numerical result ✅

C.Solve the expression for every possible variable value

D.Change all operations into addition before calculating

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Evaluating an expression means substituting the given values for the variables and then simplifying according to the correct order of operations. The goal is usually to obtain one numerical value for the specified input.

Q9. A calculator program receives the expression $2x+3y$ and the inputs $x=4$ and $y=5$ . Which process correctly represents evaluation?

A.Solve

2x+3y=0

before using the inputs

B.Replace

x

and

y

with their values, then simplify the resulting numerical expression ✅

C.Multiply

x

and

y

first because variables always have priority

D.Replace only

x

, leaving

y

unchanged

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The purpose of evaluation is to determine the value of an expression for specified inputs. Substituting $x=4$ and $y=5$ produces $2(4)+3(5)$ , which can then be simplified to a numerical result.

Q10. A student says that evaluating $4a-7$ for $a=3$ means solving $4a-7=0$ . What is the best analysis of the student's reasoning?

A.Correct, because every expression must be converted into an equation

B.Correct, because evaluation always requires finding an unknown

C.Incorrect, because evaluation substitutes the given value and simplifies rather than solving an equation ✅

D.Incorrect, because expressions cannot contain subtraction

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The student has confused evaluation with solving an equation. For $a=3$ , evaluation means replacing $a$ with $3$ , giving $4(3)-7$ , and simplifying. No unknown remains to be solved for.

Q11. A shop uses $C=50+8n$ to represent a delivery cost, where $n$ is the number of packages. The manager wants the cost for $n=6$ . Which action demonstrates evaluation?

A.Find the value of

n

that makes

C=0

B.Substitute

6

for

n

and simplify

50+8(6)

✅

C.Change the expression to

C=58n

D.Substitute

50

for

n

because it is the constant

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: In a model, evaluating means using the specified input and determining the corresponding output. Substituting $n=6$ into $C=50+8n$ gives $50+8(6)$ , which can then be simplified to the delivery cost.

Q12. A graph represents $f(x)=x+4$ . A student wants the output corresponding to $x=3$ . Which interpretation correctly describes evaluating the expression using the graph or formula?

A.Find the input where the graph crosses the

x

-axis

B.Use

x=3

as the input and determine the corresponding output, which is

7

✅

C.Change

x

4

because

4

is the constant

D.Solve

x+4=0

to determine the output

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Evaluating $f(x)=x+4$ at $x=3$ means finding the output associated with that input. Substitution gives $f(3)=3+4=7$ . On a graph, this corresponds to the point with coordinates $(3,7)$ .

Q13. Two students evaluate $3p^2+2$ at $p=-2$ . Student A substitutes $-2$ and gets $3(-2)^2+2=14$ . Student B writes $3(-2)^2+2=-12+2=-10$ . Which analysis is correct?

A.Student A is correct because the negative value is squared before multiplication ✅

B.Student B is correct because squaring a negative number keeps it negative

C.Both are incorrect because an expression cannot be evaluated at a negative input

D.Both are correct because

p^2

can be either positive or negative

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The negative input must be enclosed in parentheses before squaring. Since $(-2)^2=4$ , the expression becomes $3(4)+2=14$ . Student B incorrectly treats $(-2)^2$ as $-4$ , a common substitution error.

Q14. For $E(x)=x^2-4x+5$ , a student claims that evaluating $E(2)$ means finding the value of $x$ for which $E(x)=2$ . Which response best explains the distinction?

A.The student is correct because

E(2)

always means solving for

x

B.The student is incorrect because

E(2)

means using

2

as the input and finding the resulting value of the expression ✅

C.The student is correct because expressions and equations have identical meanings

D.The student is incorrect because

E(2)

cannot be calculated without another variable

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Function notation $E(2)$ specifies an input of $2$ , so evaluation requires substituting $2$ for $x$ . It does not ask for an input that produces an output of $2$ . In fact, $E(2)=4-8+5=1$ .

Q15. If $x=-3$ , which expression correctly represents substituting the given value into $5x-2$ ?

5-3-2

5(-3)-2

✅

5(-2)-3

5(3)-2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Substitution means replacing the variable $x$ with its given value, including its sign. Since $x=-3$ , the expression becomes $5(-3)-2$ . Parentheses make the negative value clear and prevent sign errors.

Q16. A temperature model is $T=2h+18$ , where $h$ represents hours after noon. What expression should be used to determine the temperature 5 hours after noon?

2+5+18

2(5)+18

✅

5(18)+2

2(5+18)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The variable $h$ represents the number of hours, so 5 must replace $h$ . Substituting gives $T=2(5)+18$ . The coefficient 2 remains attached to the substituted value, while 18 remains the constant.

Q17. A student substitutes $y=-4$ into $y^2-3y$ and writes $-4^2-3(-4)$ . What is the main problem with this substitution?

A.The coefficient 3 should also be replaced

B.The negative value should be enclosed in parentheses before being squared ✅

C.The subtraction sign should be changed to addition

D.The variable should remain in the expression

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When substituting a negative number for a variable, parentheses are important. The correct substitution is $(-4)^2-3(-4)$ . Without parentheses, $-4^2$ is interpreted differently under standard operation rules, causing a sign error.

Q18. A school models the number of points earned as $P=4b+3c$ , where $b$ is the number of basic tasks and $c$ is the number of challenging tasks. If a student completes $b=6$ and $c=4$ , which substitution correctly models the score?

4(6)+3(4)

✅

4+6+3+4

6(4)+4(3)

4(6+3)(4)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Each variable must be replaced by its corresponding value while keeping its coefficient and operation unchanged. Replacing $b$ with 6 and $c$ with 4 gives $4(6)+3(4)$ , preserving the original model.

Q19. A graph represents $y=3x-2$ . At $x=-2$ , which substitution should be made to find the graph's corresponding $y$ -value?

y=3-2-2

y=3(-2)-2

✅

y=-2(3-2)

y=3(2)-2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The graph's input is $x=-2$ , so that exact value replaces $x$ in the formula. The correct substitution is $y=3(-2)-2$ , which gives $y=-8$ . The negative sign must be retained.

Q20. Two students substitute $a=5$ into $2(a+3)^2$ . Student A writes $2(5+3)^2$ , while Student B writes $2(5)+3^2$ . Which evaluation is consistent with the original expression?

A.Student A, because the entire quantity

a+3

is squared ✅

B.Student B, because each number should be squared separately

C.Both students, because addition distributes over powers

D.Neither student, because substitution cannot be used inside parentheses

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The parentheses are part of the original expression and the entire quantity $a+3$ is squared. Replacing $a$ with 5 gives $2(5+3)^2$ . Student B changes the structure of the expression and therefore does not perform valid substitution.

Q21. For $x=2$ and $y=-3$ , a model is $M=(x-y)^2-2xy$ . Which substitution is correct and why is it important to preserve the parentheses around the negative value?

M=(2-3)^2-2(2)(3)

, because signs can be ignored

M=(2-(-3))^2-2(2)(-3)

, because both occurrences of

y

must be replaced by

-3

✅

M=(2+3)^2-2(2)(3)

, because every negative value should become positive

M=(2-(-3))^2-2(2)(3)

, because multiplication removes the negative sign

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Every occurrence of $y$ must be replaced by $-3$ , so the first part becomes $2-(-3)$ and the product becomes $2(2)(-3)$ . Preserving the negative sign is essential because it changes both the squared quantity and product term.

🔗 Related Topics (MCQs)

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←Previous Topic:Order of operations PEMDAS📝 How to evaluate algebraic expressionsCombining like terms:Next Topic→