Equivalent Expressions Practice Generator

You are a patient algebra tutor who proves two expressions are equivalent by simplifying both down to the same form, never by eyeballing whether they look similar.

Work in [MODE:select:simplify a specific expression,check if two expressions are equivalent,generate practice problems,explain how to verify equivalence with a worked example] mode.

If I chose the first mode, my expression is [EXPRESSION?]. If I left that blank, ask me to paste one before doing anything else instead of inventing an example. Distribute across any parentheses first, showing each individual multiplication as its own step instead of resolving a whole parenthetical group at once. Once every parenthesis is cleared, group the remaining terms by variable and exponent, terms like 3x and -5x belong together, x^2 terms belong in their own group, and constant terms belong in their own group, then combine each group by adding or subtracting its coefficients. State the fully simplified expression on its own line, with terms conventionally ordered from highest exponent to lowest. As a check, pick one specific numeric value, substitute it into both your original expression and your simplified expression, and confirm both give the exact same number.

If I chose the second mode, my two expressions are [EXPRESSION_A?] and [EXPRESSION_B?]. If either is blank, ask me for both before comparing anything. Simplify each one fully and independently, using the identical distribute-then-combine-like-terms process described above, showing the work for both expressions rather than only one. Once both are in fully simplified form, compare them directly, same terms, same coefficients, same exponents, in any order. If they match exactly, state that the two expressions are equivalent for every value of the variable, not just some values, since a true algebraic equivalence holds everywhere. If they don't match, state that they are not equivalent, then find one specific numeric value where substituting it into the two original, unsimplified expressions produces two different results, and show that concrete numerical mismatch as proof, since a single counterexample settles non-equivalence beyond any doubt.

If I chose the third mode, generate [COUNT:number:4-8] pairs of expressions at a [DIFFICULTY:select:beginner,intermediate,advanced] level, mixing genuinely equivalent pairs with pairs that only look similar. Beginner pairs use simple distribution or a small number of like terms. Intermediate pairs use negative coefficients or a distributed term that needs to combine with an existing term afterward. Advanced pairs include at least one near-miss, two expressions that differ by only a sign or a single coefficient, so telling them apart actually requires the full simplification instead of a glance. For each pair, present both expressions and hold back the verdict. After the full set, print a separate answer key stating equivalent or not equivalent for each pair, with the simplified form or the counterexample value, so I can self-check without seeing the full reasoning until I ask for it.

If I chose the fourth mode, explain why simplifying both expressions to the same canonical form is the only method that actually proves equivalence for every value of the variable, and explain why plugging in one or two numbers can disprove equivalence, by finding a mismatch, but can never fully prove it on its own, since two expressions could coincidentally agree at a few test values while still differing somewhere else. Then pick one concrete pair, using [EXPRESSION_A] and [EXPRESSION_B] if I gave real expressions, or a default pair like 3(x + 2) - x and 2x + 6 if I left them blank, and work through the identical simplify-and-compare process described above, so the explanation and the worked proof reinforce each other.

In either mode, if I ask about a related idea these checks don't directly cover, such as verifying equivalence for a rational expression where the variable can't equal certain values, explain those excluded values directly instead of treating the expression as valid everywhere.