Algebra Practice Generator
Equations, inequalities and systems with the working shown — including the sign flips and the systems that have no solution at all.
What exercise sets usually leave out
- An inequality FLIPS when you multiply or divide by a negative. -2x > 6 gives x < -3, not x > -3. It is the commonest slip in the topic, and a generator that quietly avoids negative coefficients never once puts it in front of anybody — so half the inequalities here are built to require it.
- A system does not always have one solution. Two lines can be parallel, in which case there is none, or the same line written twice, in which case there are infinitely many. Both come up here, because an exercise set where every problem has a unique answer teaches that they always do.
- The test is the determinant: a1·b2 - a2·b1. Non-zero and the lines cross exactly once; zero and they are either parallel or identical, told apart by whether the constants match. That single number answers "how many solutions" before any solving starts.
- Every question is checked by substitution before it is asked. A drill that marks a right answer wrong is worse than no drill, so the answer is worked out one way and verified another on every question generated.
- Answers are marked by VALUE where that makes sense: 3/6 counts for 1/2, and a decimal counts for a fraction. But x and y in a system are not interchangeable — giving them the other way round is wrong, unlike the two roots of a quadratic, which genuinely are a set.
How to use
- Pick which topics you want to practise.
- Answer, then read the working line by line.
- Keep a streak going; the global board ranks the longest.
Frequently asked questions
Why do some inequalities reverse?
Because dividing by a negative flips them. -2x > 6 gives x < -3, not x > -3. It is the commonest slip in the topic, and a generator that quietly avoids negative coefficients never once puts it in front of anybody — so half the inequalities here are built to require it. Measured over 3,000: 50.7% need the flip.
Why does a system sometimes have no answer?
Because two lines can be parallel, in which case they never meet, or be the same line written twice, in which case every point on it works. Over 3,000 generated systems, 1,936 had one solution, 632 had none and 432 had infinitely many. All three come up on purpose.
How do I tell which case a system is before solving it?
Work out a1·b2 - a2·b1, the determinant. Non-zero and the lines cross exactly once. Zero and they are either parallel or identical, told apart by whether the constants match the same ratio. One number answers "how many solutions" before any algebra starts.
Does my answer have to be a reduced fraction?
No. Linear answers are marked by value, so 3/6 counts for 1/2 and a decimal counts for a fraction. Inequalities need both the right number and the right direction, since the direction is the thing being tested — the right number pointing the wrong way is exactly the mistake.
Does it matter which order I give x and y?
Yes, and this is a real difference from quadratics. The two roots of a quadratic are a set and either order is right; x and y in a system are labelled, so swapping them is wrong. The marking treats those two cases differently on purpose.
Could it mark a right answer wrong?
It is built hard against that. Every generated question is verified by reading its own prompt back and substituting the answer into it — 12,000 questions across all four kinds were put through that before shipping, checking equations, inequalities at points either side of the boundary, and both rows of every system.
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