When solving 3x + 7 = 3x − 2, a student subtracts 3x from both sides and gets 7 = −2. What is the solution?
Ax = 0, because there is no variable left
Bx = 9, because 7 + 2 = 9
CNo solution, because 7 = −2 is a false statement
Dx = −2/7
When the variable cancels and leaves a false numerical statement (a contradiction like 7 = −2), the equation has no solution. No value of x can make both sides equal because the variable terms are identical on both sides but the constants differ. The common error is writing x = 0, but x has already been eliminated — there is no x left to solve for. The correct answer is 'no solution.'
Question 2 Multiple Choice
After simplifying 2(x + 4) = 2x + 8, a student arrives at 8 = 8. What is the solution?
Ax = 8
Bx = 0
CNo solution
DAll real numbers
When the variable cancels and leaves a true statement (an identity like 8 = 8), the equation is satisfied for every real number. Substituting any value of x will make both sides equal because the two expressions are algebraically identical. This is the opposite of the no-solution case: instead of a false statement, you get a statement that is always true.
Question 3 True / False
If solving an equation leads to a result like 0 = 0 after all variable terms cancel, the equation has infinitely many solutions.
TTrue
FFalse
Answer: True
A result of 0 = 0 (or any true numerical identity like 5 = 5) after eliminating variables means the two sides of the original equation are equivalent expressions. Since both sides are equal for every value of x, every real number is a solution. This is called an identity equation.
Question 4 True / False
When solving 2x + 1 = 2x + 5, the result 1 = 5 means x = 0 is the solution.
TTrue
FFalse
Answer: False
After subtracting 2x from both sides, x no longer appears in the equation — there is nothing left to solve for. The statement 1 = 5 is always false, regardless of x, which means no value of x satisfies the original equation. The correct conclusion is 'no solution,' not x = 0. Writing x = 0 confuses 'x has disappeared' with 'x equals zero.'
Question 5 Short Answer
Explain why 3x + 4 = 3x + 9 has no solution, while 3(x + 4) = 3x + 12 has infinitely many solutions.
Think about your answer, then reveal below.
Model answer: In the first equation, subtracting 3x gives 4 = 9, which is always false — no value of x can satisfy it. In the second, distributing gives 3x + 12 = 3x + 12; subtracting 3x gives 12 = 12, which is always true. The key is whether the variable terms cancel to reveal a contradiction (false statement → no solution) or an identity (true statement → all real numbers).
These two cases hinge on what remains after the variable is eliminated. When the constants on both sides differ (4 ≠ 9), the equation demands the impossible. When the expressions are algebraically identical after simplification (both sides are the same expression), the equation is trivially satisfied by everything. Recognizing which case you're in is the core skill this topic introduces.