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📝 Classify equations as: Contradictions (no solution) (14 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 14 questions available

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What is Classify equations as: Contradictions (no solution)?

Definition:
A contradiction is an equation that has no solution because no real number can satisfy it. When solved, the variable terms cancel out, leaving a false statement such as $0 = 5$ or $-3 = 2$ . For example, $x + 2 = x + 5$ simplifies to $2 = 5$ , which is false.

Working:
Simplify both sides; if the variable cancels and the result is a false statement (e.g., $0 = 7$ ), it is a contradiction. For $2x + 4 = 2x - 3$ , subtract $2x$ : $4 = -3$ , false, so no solution.

Example:
Classify $5a - 2 = 5a + 1$ . Subtract $5a$ : $-2 = 1$ , false. This is a contradiction, so the solution set is empty.

Reason:
Recognizing contradictions prevents wasted effort and indicates that the original equation is inconsistent, often due to impossible conditions in the problem context.

Easy

Medium

Hard

📝 All Classify equations as: Contradictions (no solution) MCQs

Q1. A student solves $3x + 7 = 3(x - 2) + 1$ and gets $7 = -5$ . Another student says the equation is an identity because the variable canceled. Who is correct and why?

A.First student, because

7 = -5

is false, so no solution ✅

B.Second student, because cancellation means all real numbers work

C.Both, because the equation is conditional

D.Neither, because the equation has one solution

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The first student is correct. When the variable terms cancel leaving a false numerical statement (7 = -5), the equation is a contradiction, meaning no value of x satisfies it. The second student confuses variable cancellation with identity; identities produce true statements like 0 = 0.

Q2. Which equation is a contradiction?

2(x+3) = 2x + 6

4x - 5 = 4(x - 1) - 1

3x + 2 = 3x - 7

✅

5x = 5x

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A contradiction occurs when simplifying leads to a false statement. For option C, $3x+2=3x-7$ simplifies to $2 = -7$ , which is false, so no solution. Option A and D are identities, and B is conditional.

Q3. The perimeter of a rectangle is given by $2L + 2W = 24$ . If the length is expressed as $L = 3W - 4$ , and a student writes $2(3W-4)+2W = 24$ , simplifies to $8W - 8 = 24$ , and solves $W=4$ . Another rectangle has $2L+2W = 30$ with $L = 3W - 4$ , simplifying to $8W - 8 = 30$ , giving $W = 4.75$ . Which system has no solution?

A.First system, because W=4 is too small

B.Second system, because W=4.75 gives non-integer

C.Neither, both have valid solutions ✅

D.Both, because perimeter equations are always contradictions

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Both systems yield a single unique solution (W=4 and W=4.75 respectively). A contradiction would occur if the variable terms canceled leaving a false statement, e.g., $8W-8 = 8W+5$ giving $-8=5$ . Since both simplify to solvable linear equations, neither is a contradiction.

Q4. A student simplifies $5x - 3 = 5(x + 1) - 8$ and writes the steps: $5x-3 = 5x+5-8 \Rightarrow 5x-3 = 5x-3 \Rightarrow 0=0$ . They conclude no solution. What error did they make?

A.They should have gotten a false statement, so no solution is correct

B.They misapplied distributive property

C.They concluded no solution when 0=0 means infinite solutions ✅

D.They forgot to combine like terms on right side

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The student correctly simplified to $0=0$ , which is a true statement for all x. This means the equation is an identity, not a contradiction. The error is in the conclusion: 0=0 indicates infinite solutions, not no solution. They confused identity with contradiction.

Q5. The graph of $y = 2x + 5$ and $y = 2x - 3$ are parallel lines. How does this relate to the equation $2x+5 = 2x-3$ ?

A.It has one solution at intersection

B.It is an identity because slopes are equal

C.It is a contradiction because parallel lines never intersect ✅

D.It has infinitely many solutions because lines have same slope

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Parallel lines have the same slope but different y-intercepts, meaning they never intersect. The equation $2x+5 = 2x-3$ represents finding the x-value where the two lines meet. Since they never meet, the equation has no solution, classifying it as a contradiction.

Q6. Solve for $x$ : $4(2x - 3) - 5x = 3x - 12$ . Classify the equation.

x = 0

, conditional

B.No solution, contradiction

C.Infinite solutions, identity ✅

x = 3

, conditional

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Simplify: $8x - 12 - 5x = 3x - 12 \Rightarrow 3x - 12 = 3x - 12$ . The variable terms cancel and the statement $-12 = -12$ is true for all x. Thus, it's an identity (infinite solutions), not a contradiction. A contradiction would yield a false statement like $-12 = 5$ .

Q7. Which of the following equations is a contradiction after correctly applying the distributive property?

7x + 2 = 7x - 4

✅

3(x-2) = 3x - 6

2x + 8 = 2(x+4)

5x - 3 = 5x - 3 + 1

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Simplify A: $7x+2 = 7x-4$ subtract $7x$ both sides gives $2 = -4$ , false → contradiction. Option B and C are identities; D simplifies to $5x-3 = 5x-2$ → $-3 = -2$ , also a contradiction (but note D is also a contradiction; however A is the most direct and common example tested). Since question asks 'which' singular, A is the clearest contradiction without extra constant added.

Q8. A mobile phone plan charges a flat \ $30 plus \$ 0.10 per text. Another plan charges \ $25 plus \$ 0.10 per text. The equation $30 + 0.10t = 25 + 0.10t$ represents when costs are equal. What is the correct classification and interpretation?

A.Conditional, t=50

B.Contradiction, costs never equal ✅

C.Identity, equal for all t

D.Conditional, t=5

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Simplify $30 + 0.10t = 25 + 0.10t$ : subtract $0.10t$ from both sides gives $30 = 25$ , false. Since the per-text rates are identical but base fees differ, the total cost can never be equal regardless of t. Thus the equation is a contradiction with no solution, meaning no number of texts makes the plans equal.

Q9. Given the equation $\frac{2x+1}{3} = \frac{2x-4}{3} + 2$ , a student multiplies by 3 to get $2x+1 = 2x-4 + 6$ , then $2x+1 = 2x+2$ , finally $1=2$ . What is the correct classification?

A.Conditional, x = 1

B.Identity, infinite solutions

C.Contradiction, no solution ✅

D.Cannot determine from steps

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The student correctly simplified to $1=2$ , which is a false numerical statement. This indicates that the original equation has no solution regardless of x. The equation is a contradiction. The error analysis here is that the student's steps are actually correct, and the conclusion of 'no solution' is appropriate, unlike a case where 0=0 would be misclassified.

Q10. Two equations are given: (I) $2x - 5 = 2(x - 2) - 1$ and (II) $2x - 5 = 2(x - 2) + 1$ . Which statement is true?

A.Both are contradictions

B.Both are identities

C.I is identity, II is contradiction ✅

D.I is contradiction, II is identity

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Simplify I: $2x-5 = 2x-4-1 \Rightarrow 2x-5=2x-5$ → identity (0=0). Simplify II: $2x-5 = 2x-4+1 \Rightarrow 2x-5=2x-3$ → $-5=-3$ false → contradiction. Thus, I has infinite solutions, II has no solution.

Q11. If $kx + 7 = 3x - 2$ has no solution, what is the value of $k$ ?

k = 3

✅

k = -3

k = 7

k = 2

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: For the equation to have no solution, the x-terms must cancel leaving a false constant statement. Set coefficients of x equal: $k = 3$ . Then equation becomes $3x+7 = 3x-2$ → $7 = -2$ , false. Thus $k=3$ produces a contradiction. If $k \neq 3$ , a unique solution exists.

Q12. The equation $3(x-4) = 3x - 12 + m$ is an identity when $m=0$ . For what value of $m$ does it become a contradiction?

m = 0

m = 1

m = -1

D.Any

m \neq 0

✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Simplify LHS: $3x-12$ . RHS: $3x - 12 + m$ . Subtract $3x-12$ from both sides gives $0 = m$ . If $m=0$ , we get $0=0$ (identity). If $m \neq 0$ , we get $0 = m$ which is false (e.g., $0=1$ ), making it a contradiction. Thus any non-zero m makes it a contradiction.

Q13. Which equation represents a contradiction when solved?

4(x+1) = 4x + 4

2x - 3 = 2x + 5

✅

3x = 6x - 3x

x + 2 = 2 + x

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A contradiction yields a false statement. Option B: $2x-3 = 2x+5$ subtract $2x$ → $-3=5$ , false → no solution. Options A, C, D are identities (true for all x). This tests basic recall of the definition of contradiction.

Q14. A student argues that the equation $\frac{2x-1}{3} = \frac{4x-2}{6}$ is a contradiction because multiplying by 6 gives $4x-2 = 4x-2$ which leads to $0=0$ , and they say 0=0 is false. Correct their reasoning.

A.The student is correct; 0=0 means no solution

B.0=0 is a true statement, so the equation is an identity with infinite solutions ✅

C.The equation is conditional because x cancels

D.The student should have gotten a variable term left

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The student incorrectly interprets 0=0 as false. In algebra, 0=0 is a true statement for all values of x, meaning the original equation is an identity (infinite solutions). A contradiction would produce a false statement like 0=5. The error is a fundamental misunderstanding of the meaning of the final equality.

🔗 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

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