Linear Equations Worksheet Generator
Every problem the generator produces has an integer solution, so students can self-check. Pick a difficulty and a count and the generator builds a fresh set — or pass a seed to regenerate the same sheet on another device.
Built backwards from the answer
A practice sheet whose answers come out as awkward fractions teaches the wrong lesson. The student who reaches minus seven thirds cannot tell whether the method was right and the arithmetic ugly, or whether something went wrong three lines earlier — and that uncertainty is what makes self-checking impossible.
This generator avoids the problem by construction. Every equation it produces has a whole-number solution, so an answer that is not an integer is a signal to look again before turning to the key.
The generator does not invent an equation and then hope it solves cleanly. It picks the solution first — a whole number, never zero — and then builds an equation around it, computing whatever right-hand side makes that value correct.
Working in that direction guarantees the property that matters without any filtering or retrying. It also means the answer key costs nothing to produce: the answer was known before the question existed.
How to use this calculator
- Choose how many problems Between 1 and 30. A larger number is clamped down rather than refused, and an entry that is not a number falls back to ten.
- Pick the difficulty Easy gives x + b = c, medium ax + b = c, and hard a(x + b) = cx + d. Medium is the default here, since it is the shape most one-variable practice targets.
- Set a seed, or leave it blank A whole number reproduces the same sheet anywhere. Left blank, one is chosen at random and printed with the worksheet so the sheet stays reproducible after the fact.
- Work the sheet, then check the key Problems come first and the key last, so the page can be folded or cut between them for self-marking.
How each equation is built
The seed is read first and initialises a mulberry32 generator; a blank or unreadable entry gets a random seed, which is reported alongside the sheet. Every draw that follows comes from that one generator, which is what makes a seeded sheet identical everywhere.
At easy level the solution is drawn as a non-zero integer between −9 and 9, a non-zero constant is drawn from the same range, and the right-hand side is simply their sum. The equation is then printed with that constant and that total, and it necessarily solves to the value drawn first.
Medium adds a coefficient between 2 and 6 in front of the variable. The right-hand side becomes the coefficient times the solution, plus the constant — so the division at the end of the student's work always comes out exactly, because the numerator was constructed as a multiple of the divisor.
Hard puts the variable on both sides, as a bracket on the left against a variable term on the right. The inner coefficient is drawn strictly below the outer one, from 1 up to one less than it, so collecting the variable terms always leaves a positive coefficient of at least one. The right-hand constant is then whatever makes the drawn solution satisfy the equation. That constraint is why a hard problem never collapses to a negative or zero coefficient after collection.
What each input means
- n Number of problems — form field “Number of problems”
- How many equations appear on the sheet, clamped to between 1 and 30. Units: problems.
- d Difficulty — form field “Difficulty”
- Easy, medium or hard, selecting the shape of the equation rather than only the size of its numbers.
- s Seed — form field “Seed (optional)”
- Optional whole number fixing the whole sheet. Always reported with the output, whether supplied or generated.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
Ten easy equations
Ten problems at the easy level from a fixed seed. Each is a single addition or subtraction to undo.
Inputs Number of problems = 10, Difficulty = easy, Seed (optional) = 20260820
- Seed 20260820 (use this seed to regenerate the same worksheet)
- Count 10 problems
- Problem 1. Solve: x + 8 = 7
- Problem 2. Solve: x + 8 = 9
- Problem 3. Solve: x + 4 = -4
- Problem 4. Solve: x − 8 = -16
- Problem 5. Solve: x + 1 = 0
- Problem 6. Solve: x − 8 = -11
- Problem 7. Solve: x − 3 = 1
- Problem 8. Solve: x + 7 = -1
- Problem 9. Solve: x − 9 = -3
- Problem 10. Solve: x + 5 = 8
- Answer key 1. x = -1; 2. x = 1; 3. x = -8; 4. x = -8; 5. x = -1; 6. x = -3; 7. x = 4; 8. x = -8; 9. x = 6; 10. x = 3
Result 10-problem worksheet — Linear equations (easy) (seed 20260820)
There is no coefficient on the variable, so each problem takes one step: move the constant across. Roughly half the constants are negative, which is where the useful practice lies — subtracting a negative rather than the arithmetic itself.
No solution is ever zero. That is deliberate: an answer of zero makes several different mistakes produce the same right-looking result, which defeats self-checking.
Fifteen hard equations, seeded
Fifteen problems with the variable on both sides, from the seed 42. This is the shape that needs collecting before isolating.
Inputs Number of problems = 15, Difficulty = hard, Seed (optional) = 42
- Seed 42 (use this seed to regenerate the same worksheet)
- Count 15 problems
- Problem 1. Solve: 3(x + 3) = 2x + 10
- Problem 2. Solve: 4(x − 2) = 2x − 16
- Problem 3. Solve: 3(x − 2) = 2x − 1
- Problem 4. Solve: 3(x − 3) = 2x − 6
- Problem 5. Solve: 4(x − 4) = 2x − 12
- Problem 6. Solve: 2(x + 1) = x + 6
- Problem 7. Solve: 2(x − 3) = x − 9
- Problem 8. Solve: 2(x + 3) = x + 0
- Problem 9. Solve: 3(x − 4) = x − 4
- Problem 10. Solve: 4(x − 2) = 2x − 6
- Problem 11. Solve: 3(x + 2) = 2x + 2
- Problem 12. Solve: 3(x − 2) = x − 14
- Problem 13. Solve: 4(x + 2) = 3x + 10
- Problem 14. Solve: 5(x − 1) = 4x − 10
- Problem 15. Solve: 2(x − 4) = x − 13
- Answer key 1. x = 1; 2. x = -4; 3. x = 5; 4. x = 3; 5. x = 2; 6. x = 4; 7. x = -3; 8. x = -6; 9. x = 4; 10. x = 1; 11. x = -4; 12. x = -4; 13. x = 2; 14. x = -5; 15. x = -5
Result 15-problem worksheet — Linear equations (hard) (seed 42)
Each problem needs three moves rather than one: expand the bracket, gather the variable terms on one side, then isolate. The expansion is where most errors enter, because the outer coefficient must multiply both terms inside.
The coefficient left after collecting is always positive, since the inner coefficient is drawn strictly smaller than the outer one. A negative coefficient at that stage means the collection went the wrong way.
Reading the result
A fractional answer is a signal
Every solution on every sheet is a whole number. Arriving at anything else means an error has occurred, and knowing that before consulting the key is most of the value of the guarantee.
What the difficulty actually changes
Not the size of the numbers so much as the number of steps. Easy needs one operation, medium two, and hard four including the expansion — so the levels correspond to stages in learning rather than to arithmetic strain.
The seed identifies a sheet, not a difficulty
Reproducing a worksheet needs the seed, the count and the difficulty together. Changing the level draws a different number of values per problem, so the sequence diverges immediately and the numbers share nothing with the original.
When you would use this
Homework that can be self-marked
Because every answer is an integer, a student can substitute back and confirm the equation balances. That closes the loop without the key, which is the habit worth building.
Different numbers, same task
Running the same count and difficulty under different seeds gives every student their own sheet at an identical level of difficulty — useful for retakes as much as for a class set.
Assumptions and limitations
What this calculator assumes
- Every equation has exactly one solution, and that solution is a non-zero integer.
- All coefficients and constants are integers.
- At the hard level the inner coefficient is strictly smaller than the outer one, so collecting always leaves a positive coefficient.
- The same seed, count and difficulty reproduce the same sheet exactly.
Where it stops being the right tool
- One variable only: simultaneous equations and inequalities are not generated.
- No fractional or decimal coefficients, and no equation whose solution is not an integer.
- Three fixed shapes, so a specific pattern such as a variable in a denominator cannot be requested.
- At most 30 problems per sheet, all at one difficulty.
Common mistakes
Expanding a bracket onto only the first term
Why it happens. In a(x + b) the multiplication has to reach both the variable and the constant, and the constant is the one that gets forgotten because it sits further from the coefficient.
How to avoid it. Check by substituting your answer into the original equation rather than the expanded one. A dropped product shows up immediately as an imbalance.
Collecting variable terms in the wrong direction
Why it happens. Either side can be chosen, so it is easy to subtract the larger coefficient from the smaller and end up with a negative where these problems never produce one.
How to avoid it. Move the smaller variable term across, which here is always the one on the right. The coefficient that remains is positive by construction.
Assuming a non-integer answer means a hard problem
Why it happens. Awkward numbers feel like the sheet getting harder rather than like a mistake, especially under time pressure.
How to avoid it. Treat any fraction as an error signal. Every problem on every difficulty solves to a whole number, so there is no legitimate fractional answer to reach.
Frequently asked questions
Are the solutions always integers?
Yes, on every difficulty. The generator draws the answer first as a non-zero whole number and then constructs an equation around it, so a clean solution is guaranteed rather than filtered for.
Can I make a quiz and the answer key separately?
The problems are listed first and the full key last, so the page can be cut or folded between them. Working the sheet before the key comes into view is the intended use.
How does the seed work?
It is the starting value for the pseudo-random generator that draws every number on the sheet. The same seed with the same count and difficulty rebuilds an identical worksheet on any device.
What does each difficulty look like?
Easy is x + b = c, one step. Medium is ax + b = c, adding a division. Hard is a(x + b) = cx + d, which needs expanding and collecting before the variable can be isolated.