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📝 Solve equations that require simplification (15 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 15 questions available

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What is Solve equations that require simplification?

Definition:
Solving equations that require simplification involves first simplifying each side of the equation by distributing, combining like terms, and clearing fractions or decimals before applying the properties of equality to isolate the variable. This is necessary when equations are not in their simplest form.

Working:
Start by distributing any constants over parentheses, then combine like terms on each side. If fractions exist, multiply by the LCD to clear them. Then, use addition/subtraction to move variable terms to one side and constant terms to the other, finally multiply/divide to solve. For example, solve $\frac{1}{2}x + 3 = 7$ : multiply by 2: $x + 6 = 14$ , then subtract 6: $x = 8$ .

Example:
Solve $3(x - 2) + 4 = 16$ . Distribute: $3x - 6 + 4 = 16$ , combine: $3x - 2 = 16$ , add 2: $3x = 18$ , divide by 3: $x = 6$ . Check: $3(6-2)+4 = 3(4)+4 = 12+4 = 16$ .

Reason:
Simplification reduces complexity, preventing errors and making the equation easier to solve, especially when dealing with multi-step problems.

Easy

Medium

Hard

📝 All Solve equations that require simplification MCQs

Q1. A student solves $3(x-4)+2x = 5x-12$ and writes 'All real numbers'. Is the student correct?

A.Yes, because both sides simplify to the same expression ✅

B.No, because the equation has no solution

C.Yes, but only for positive numbers

D.No, because the variable cancels leaving a false statement

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The student is correct. Simplifying the left side: $3x-12+2x = 5x-12$ , which is identical to the right side. This is an identity, true for all real numbers. The other options confuse identities with contradictions or conditional equations.

Q2. Which equation requires combining like terms on both sides before solving?

2(x+1)=8

3x-4=2x+5

5x+3x-2=4x+7

✅

x/2+3=7

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: In option C, the left side has two like terms $5x$ and $3x$ that must be combined to $8x$ before any inverse operations. Option A requires distribution, B requires variable terms on both sides, and D involves fractions. HOTS: identify the need for simplification as a prerequisite step.

Q3. A rectangle's length is $2x+3$ and width is $x-1$ . If the perimeter is 34, which equation correctly models this?

2(2x+3)+2(x-1)=34

2(2x+3+x-1)=34

✅

(2x+3)(x-1)=34

2x+3+x-1=34

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The perimeter formula is $2(\text{length}+\text{width})$ . Substituting gives $2((2x+3)+(x-1))=34$ , which simplifies to $2(3x+2)=34$ . Option A incorrectly applies the factor 2 separately, C uses area, and D ignores the factor 2 entirely. This requires modeling a real-world scenario.

Q4. What is the first step to solve $4(2x-5)-3x = 2(x+1)+10$ ?

A.Add

3x

to both sides

B.Distribute on both sides ✅

C.Subtract 10 from both sides

D.Combine

2x

and

x

on the right

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Distributing is the first necessary step to remove parentheses on both sides: $8x-20-3x = 2x+2+10$ . Without distribution, simplification is impossible. The other options are valid later steps but cannot be performed correctly without first distributing. This tests the hierarchy of operations in equation solving.

Q5. A student solved $5(x-2)=3x+4$ and got $x=7$ . When checking, $15=15$ . What does this verify?

A.The solution is correct ✅

B.The solution is incorrect but check was wrong

C.The solution is

x=14

D.There is no solution

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Checking: $5(7-2)=5(5)=25$ and $3(7)+4=21+4=25$ . The solution $x=7$ satisfies the equation. This question requires performing the check and interpreting its meaning. Distractors include common arithmetic errors or misinterpreting the check as proving identity.

Q6. Two students solve $2(3x-1)+4=6x+2$ . One says 'No solution', other says 'All real numbers'. Who is correct?

A.First student

B.Second student ✅

C.Both are wrong

D.Both are correct depending on domain

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Simplify LHS: $6x-2+4=6x+2$ , RHS is $6x+2$ . The equation is an identity, so all real numbers work. The first student likely subtracted $6x$ and got $-2+4=2$ then incorrectly concluded no solution. This distractor tests ability to recognize identities versus contradictions.

Q7. Which graph represents the solution to $2(x+1)-3=5$ ?

A.A single point at

x=3

✅

B.A single point at

x=2

C.A horizontal line

y=5

D.All points on the number line

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Solving: $2x+2-3=5 \Rightarrow 2x-1=5 \Rightarrow 2x=6 \Rightarrow x=3$ . The solution is a single point on the number line. Option B is a common arithmetic error (ignoring the $-1$ ), C confuses equation with function, D confuses with identity. Graph interpretation requires connecting algebraic solution to visual representation.

Q8. If $3(2x-1)-2(1-x)=0$ , which of these is equivalent after simplifying?

6x-3-2+2x=0

✅

6x-3-2-2x=0

6x-3-2+x=0

3x-1-2+2x=0

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Correct distribution: $3(2x-1)=6x-3$ and $-2(1-x)=-2+2x$ . So LHS = $6x-3-2+2x$ . Option B has wrong sign on $2x$ , C has wrong coefficient on x, and D didn't distribute correctly. This tests careful distribution with a negative sign, a common source of errors.

Q9. A car rental charges $25+0.15m$ dollars, another charges $40+0.05m$ . For what mileage $m$ are costs equal?

m=150

✅

m=100

m=200

m=300

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Set $25+0.15m=40+0.05m$ . Simplifying: $0.15m-0.05m=40-25 \Rightarrow 0.10m=15 \Rightarrow m=150$ . This models a real decision problem. Distractors come from arithmetic misplacements or ignoring constant terms. Students must translate words to equation and solve with simplification.

Q10. What is the solution to $\frac{2x-1}{3}+\frac{x+2}{2}=1$ ?

x=1

x=7/5

✅

x=2/7

D.No solution

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Multiply by LCD 6: $2(2x-1)+3(x+2)=6 \Rightarrow 4x-2+3x+6=6 \Rightarrow 7x+4=6 \Rightarrow 7x=2 \Rightarrow x=2/7$ . Wait, recalc: $4x+3x=7x$ , $-2+6=4$ , so $7x+4=6 \Rightarrow 7x=2 \Rightarrow x=2/7$ . Option C is correct. Distractors include wrong LCD or sign errors.

Q11. Given the equation $4x-3=2x+5$ , which step would create an equivalent equation but not change the solution set?

A.Add

3x

to both sides ✅

B.Multiply both sides by 0

C.Divide both sides by

x-1

D.Square both sides

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Adding $3x$ to both sides is a valid reversible operation that preserves equivalence. Multiplying by zero loses information, dividing by an expression that might be zero risks losing roots, and squaring can introduce extraneous solutions. This tests understanding of equivalence-preserving operations beyond rote procedures.

Q12. A student's work: $5x-2=3x+10 \Rightarrow 2x=12 \Rightarrow x=6$ . Which property was used to get from first to second step?

A.Subtraction property ✅

B.Addition property

C.Division property

D.Distributive property

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: To get $2x=12$ , the student subtracted $3x$ from both sides (or added $-3x$ ), which is the subtraction property of equality. The addition property applies to adding the same number, division to multiplying by reciprocal, and distribution is for parentheses. This tests precise vocabulary and recognition of algebraic properties.

Q13. The equation $2(3x-4)=6x-8$ has how many solutions?

A.0

B.1

C.2

D.Infinitely many ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Simplifying LHS: $6x-8$ , RHS is $6x-8$ . Both sides are identical, so it's an identity. A student might incorrectly solve to get $0=0$ and think no solution (Option A), or stop early and pick 1 (Option B). This tests recognition of identities and the meaning of $0=0$ versus $0=5$ .

Q14. Compare two methods: Method A simplifies first, then solves. Method B solves directly. Which is better for $2(x+3)+4x=3(2x-1)+9$ ?

A.Method A, because simplification prevents errors

B.Method B, because it's faster

C.Both are equivalent and yield all real numbers ✅

D.Neither works because equation has no solution

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Using Method A: LHS = $2x+6+4x=6x+6$ , RHS = $6x-3+9=6x+6$ . Identity, all real. Method B: subtract $6x$ from both sides gives $6=6$ , also identity. Both methods are valid and equivalent. This question compares strategies and requires recognizing identity regardless of method.

Q15. For the equation $3(2x-5)-4 = 2(3x-5)-9$ , a student writes 'No solution'. A classmate says 'There is a solution'. Who is correct?

A.Student is correct ✅

B.Classmate is correct

C.Both are correct for different domains

D.Neither, the equation is an identity

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Simplify LHS: $6x-15-4=6x-19$ . RHS: $6x-10-9=6x-19$ . Both sides identical, so it's an identity, not no solution. The student likely subtracted $6x$ and got $-19=-19$ then mistakenly concluded no solution (confusing with $-19=20$ ). The classmate is wrong too. Correct answer is 'Neither, the equation is an identity'. This is HOTS error analysis.

🔗 Related Topics (MCQs)

📝 Subtraction and Addition Properties of Equality MCQs 📝 How to verify a solution to an equation MCQs 📝 Subtraction Property of Equality MCQs 📝 Addition Property of Equality MCQs 📝 Solving equations with simplification MCQs

←Previous Topic:Solve equations with variables divided by a constant📝 Solve equations that require simplificationTranslate sentences to equations and solve:Next Topic→