📝 Evaluate Variable Expressions with Integers (14 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 14 questions available

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What is Evaluate Variable Expressions with Integers?

Definition:
Evaluating variable expressions with integers involves substituting given integer values for the variables and then simplifying the resulting numerical expression by performing all operations with signed numbers according to the order of operations to find the final value.

Working:
Replace each variable with its assigned integer, then apply PEMDAS: simplify inside parentheses, evaluate exponents, then multiply/divide from left to right, and finally add/subtract from left to right, using integer arithmetic rules for each step.

Example:
Evaluate $3x - 2y$ when $x = -4$ and $y = 5$ .
Solution: Substitute $x=-4, y=5$ : $3(-4) - 2(5) = -12 - 10 = -22$ .

Reason:
This skill allows us to compute outputs of functions, check solutions of equations, and apply formulas in contexts like physics or finance, where variables represent measurable quantities with signed values.

Easy

Medium

Hard

📝 All Evaluate Variable Expressions with Integers MCQs

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

A.-11 ✅

B.-5

C.5

D.11

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Substitute $x=-4$ and $y=3$ carefully into the expression. This gives $2(-4)-3=-8-3=-11$ . The key reasoning is to preserve the negative sign on $x$ while multiplying before performing the subtraction.

Q2. A temperature model is given by $T=3n-8$ , where $n$ is the number of hours after midnight. What temperature does the model predict when $n=-2$ ?

A.-14 ✅

B.-2

C.2

D.14

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Replacing $n$ with $-2$ gives $T=3(-2)-8$ . Multiplication produces $-6$ , and then $-6-8=-14$ . A common mistake is treating the negative input as positive or incorrectly changing subtraction into addition.

Q3. Two students evaluate $4a-2b$ for $a=-3$ and $b=-5$ . Student 1 gets $-2$ , while Student 2 gets $-22$ . Which evaluation is correct, and why?

A.Student 1, because both terms should remain negative

B.Student 2, because

4(-3)-2(-5)=-12+10=-2

C.Student 1, because

4(-3)-2(-5)=-12-10=-22

D.Student 2, because subtracting a negative changes the second term to positive ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The correct substitution is $4(-3)-2(-5)$ . This becomes $-12+10=-2$ , so Student 1 is correct. The crucial idea is that subtracting the negative quantity $2(-5)$ is equivalent to adding $10$ , not subtracting $10$ .

Q4. A game score is modeled by $S=5p-3q+10$ . A player earns $p=-2$ penalty points and $q=-4$ bonus adjustments. What is the resulting score?

A.0

B.12 ✅

C.20

D.32

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Substitute both negative values: $S=5(-2)-3(-4)+10$ . This becomes $-10+12+10=12$ . The realistic trap is forgetting that $-3(-4)$ is positive, which would incorrectly reduce the score instead of increasing it.

Q5. A graph of $y=2x-5$ contains the plotted points $(-2,-9)$ , $(0,-5)$ , $(2,-1)$ , and $(4,3)$ . If the graph is used to evaluate the expression at $x=4$ , what value should be obtained?

A.-13

B.-3

C.3 ✅

D.13

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: At $x=4$ , the corresponding plotted point is $(4,3)$ , so the expression evaluates to $3$ . Algebraically, $2(4)-5=8-5=3$ , confirming the graph interpretation and providing an independent check.

Q6. For $m=-6$ , compare the expressions $A=3m+8$ and $B=2(m-1)$ . Which statement correctly compares their values?

A=B=-10

A=-10

and

B=-14

, so

A>B

✅

A=-26

and

B=-14

, so

A<B

A=-10

and

B=14

, so

A<B

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Evaluate each expression separately. $A=3(-6)+8=-18+8=-10$ , while $B=2(-6-1)=2(-7)=-14$ . Since $-10$ is greater than $-14$ , $A>B$ . This requires careful substitution and comparison of negative integers.

Q7. An integer $k$ satisfies $k<0$ . A student claims that $3k-2$ must be positive because multiplying by $3$ makes the number larger. Which conclusion is always true?

3k-2>0

3k-2=0

3k-2<0

✅

D.Its sign cannot be determined

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Because $k<0$ , multiplying by the positive number $3$ keeps $3k$ negative. Subtracting $2$ makes the result even smaller, so $3k-2<0$ for every negative integer $k$ . The student's error confuses magnitude with sign.

Q8. If $x=-5$ , what is the value of $3x+7$ ?

A.-22

B.-8 ✅

C.8

D.22

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Substitute $x=-5$ into the expression to get $3(-5)+7$ . Multiplication gives $-15$ , and adding $7$ results in $-8$ . The negative sign belongs to the substituted value and must be retained throughout the calculation.

Q9. A student evaluates $2a^2-3a$ when $a=-4$ and writes $-44$ . Which value is correct, and what mistake most likely caused the student's answer?

A.20; the student squared

-4

correctly but mishandled the second term ✅

B.44; the student treated

(-4)^2

-16

C.44; the student changed

-3(-4)

into

-12

D.-20; the student should have kept both substituted terms negative

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substitution gives $2(-4)^2-3(-4)=2(16)+12=44$ , so the listed value in option A is not numerically correct. Therefore this item requires correction.

Q10. A delivery company models its daily balance with $B=120+5x-2y$ . On one day, $x=-6$ represents penalties and $y=-4$ represents recovered costs. What balance does the model predict?

A.74

B.82

C.98 ✅

D.164

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Substitute $x=-6$ and $y=-4$ : $B=120+5(-6)-2(-4)$ . This becomes $120-30+8=98$ . The important reasoning step is recognizing that subtracting $2(-4)$ increases the balance by $8$ .

Q11. Two students evaluate $4m-3n+2$ for $m=-3$ and $n=-2$ . Student A obtains $-16$ , while Student B obtains $-4$ . Which student is correct?

A.Student A, because both substituted terms are negative

B.Student B, because

4(-3)-3(-2)+2=-12+6+2=-4

✅

C.Both students, because negative substitution can produce two values

D.Neither student, because negative values cannot be substituted into this expression

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Substitution gives $4(-3)-3(-2)+2=-12+6+2=-4$ . Student B correctly recognizes that subtracting a negative quantity changes that part of the calculation to addition. Student A incorrectly treats both terms as negative.

Q12. A graph represents $y=-2x+1$ . The plotted points include $(-3,7)$ , $(-1,3)$ , $(1,-1)$ , and $(3,-5)$ . Which point confirms the correct evaluation when $x=-3$ ?

(-3,-5)

(-3,-7)

(-3,7)

✅

(7,-3)

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Substituting $x=-3$ gives $y=-2(-3)+1=6+1=7$ . Therefore the graph should contain $(-3,7)$ . This connects symbolic substitution with graphical interpretation and helps verify whether the negative input was handled correctly.

Q13. A machine's output is modeled by $P=2r^2-5r-3$ . During testing, $r=-2$ . What is the predicted output, and which calculation best justifies it?

A.5, because

2(-2)^2-5(-2)-3=8+10-3

B.15, because

2(-2)^2-5(-2)-3=8+10-3=15

✅

C.-15, because the squared negative value remains negative

D.-5, because every term containing a negative input must remain negative

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Substituting $r=-2$ gives $2(-2)^2-5(-2)-3$ . Since $(-2)^2=4$ , the expression becomes $8+10-3=15$ . The strongest reasoning separates exponentiation from the signs of the other terms.

Q14. For negative integers $p$ and $q$ , a student claims that $p^2+q$ must always be negative because $q$ is negative. Which statement best evaluates the claim?

A.The claim is always true because both variables are negative

B.The claim is always false because

p^2

is always negative

C.The claim is not always true because

p^2

is positive and may be larger than

|q|

✅

D.The claim cannot be evaluated without knowing whether

p=q

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: When $p$ is negative, $p^2$ is positive, while $q$ remains negative. Their sum depends on their magnitudes. For example, $p=-5$ and $q=-3$ gives $25-3=22$ , disproving the claim that the result must always be negative.

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