Solve a system of two or three linear equations step by step with Cramer's rule, elimination, substitution or Gaussian elimination, and get exact fractions, the type of system and a check.
System of Equations Calculator - Steps, 2 and 3 Unknowns
Solve a system of two or three linear equations step by step with Cramer's rule, elimination, substitution or Gaussian elimination, and get exact fractions, the type of system and a check.
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System of equations calculator that keeps the fractions exact
Two or three linear equations go in exactly as they appear in the textbook, unknowns on either side, parentheses and fractions included, and out come x, y (and z) as exact fractions with every step of the method your teacher asked for. For 2x + 3y = 7 and x - y = 1 the answer is x = 2, y = 1, the main determinant is D = -5, and both equations are checked by substitution.
Solved on paper
- Rearrange 3(x - 1) = 2y before you can start
- Clear fractions such as 1/2x + y/3 = 2 by hand
- Decimals like 0.333 creep in and the check never quite comes out even
- A determinant of 0 leaves you guessing between no solution and infinitely many
Solved here
- Standard form shown first: 3x - 2y = 3
- Answers as fractions, x = 7/2, y = 3/4, plus decimals
- Every equation substituted back, row by row
- The type of system named: one solution, none, or a general solution such as x = 2 - y
From the worksheet to x, y and z in five moves
- Number of equations - two equations for x and y, or three for x, y and z. The third box appears only for the 3 × 3 system.
- Solution method - Cramer's rule, elimination (addition), substitution or Gaussian elimination. The answer is the same; only the written steps change. Left empty, the steps use Cramer's rule.
- Equation 1 - typed the way it is printed, for example 2x + 3y = 7 or 3(x - 1) = 2y. Multiplication can be implied: 2x means 2 times x.
- Equation 2 (and Equation 3) - same rules; an unknown that does not appear in an equation is simply left out.
- Read the result - the solution tile, the type of system, the main determinant D, the steps and the check table.
Decimals take a point, 0.5x, and numbers go in without thousands separators. Fractions can be written as 1/2x or x/2; both mean one half of x.
One system, four methods side by side
The table runs the same system, 2x + 3y = 7 and x - y = 1, through each method the calculator can write out. All four land on x = 2, y = 1; what differs is the work on the page and the kind of system each method suits.
| Method | What the steps show | For this system | Works best when |
|---|---|---|---|
| Cramer's rule | D, then Dx and Dy with one column replaced by the constants | D = -5, Dx = -10, Dy = -5, so x = 2, y = 1 | You need a formula answer or the determinant itself |
| Elimination (addition) | Multipliers that make one unknown's coefficients opposites, then the sum of the equations | Equation 2 times 3, add: 5x = 10 | Coefficients are whole numbers with an easy common multiple |
| Substitution | One unknown solved from one equation and put into the other | x = 1 + y, then 5y = 5 | An unknown already has a coefficient of 1 or -1 |
| Gaussian elimination | Augmented matrix and row operations R2 := R2 - k · R1 down to the identity matrix | Matrix [2 3 | 7; 1 -1 | 1] reduced to [1 0 | 2; 0 1 | 1] | Three unknowns, or the determinant is 0 |
The elimination row above describes the textbook move; the calculator's own steps pick the multipliers from the least common multiple of the y coefficients, so on screen equation 2 is multiplied by -3 when the two y terms carry the same sign. With three equations, elimination and substitution have no single fixed pattern, so the calculator writes those requests out as Gaussian elimination, which is the same adding of multiples, done column by column.
Before and after: six systems as typed and as solved
Each card shows a system the way it was entered and what the calculator returned. The numbers are copied from its output.
After: equation 1 multiplied by 6 gives 3x + 2y = 12, equation 2 multiplied by 2 gives x - 2y = 2; adding them gives 4x = 14.
Result: x = 7/2 = 3.5, y = 3/4 = 0.75, D = -2/3.
After: standard form 3x - 2y = 3; D = 5, Dx = 15, Dy = 15.
Result: x = 3, y = 3.
After: x = 50 - y, then -5y = -125.
Result: 25 adult and 25 child tickets.
After: equation 2 times 20 gives 2x + 5y = 120; equation 1 times 5, equation 2 times -1, sum 3x = 30.
Result: 10 L of the 10% solution and 20 L of the 25% one.
After: D = 7, Dx = 7, Dy = 14, Dz = 21.
Result: x = 1, y = 2, z = 3, all three equations hold.
After: three pivot columns, row operations such as R3 := R3 - 3 · R1, D = -10.
Result: x = 6/5 = 1.2, y = 0, z = 7/5 = 1.4.
One solution, none, or infinitely many: what D tells you first
The main determinant D is built from the coefficients only. Its value is the quickest signal of which of the three outcomes you are looking at, and the table shows where it stops being enough. Every row is a real run of the calculator.
| Determinants | Type of system | Example | Calculator's answer |
|---|---|---|---|
| D ≠ 0 | Consistent, independent | 2x + 3y = 7, x - y = 1 | x = 2, y = 1 |
| D = 0, some Dx, Dy ≠ 0 | Inconsistent | x + y = 2, 2x + 2y = 5 | No solution, Dx = -1, Dy = 1 |
| D = 0, all others 0 | Consistent, dependent | x + y = 2, 2x + 2y = 4 | x = 2 - y, y any real number |
| D = 0, all others 0 | Inconsistent | x + y + z = 1, x + y + z = 2, x + y + z = 3 | No solution, rank 1 |
| D = 0, Dx = 2 | Inconsistent | x + y + z = 1, x + 2y + 3z = 4, 2x + 3y + 4z = 7 | No solution, rank 2 |
| D = 0, all others 0 | Consistent, dependent | x + y + z = 1, x + 2y + 3z = 4, 2x + 3y + 4z = 5 | x = -2 + z, y = 3 - 2z |
The fourth row is the trap. Three parallel planes give D = 0 and every replaced determinant equal to 0 as well, which looks like the dependent case, yet the equations contradict each other. That is why the calculator never stops at the determinants when D and all the replaced determinants are 0: it hands the system to Gaussian elimination, which ends either in a row such as 0 = 1 or in a general solution.
How much arithmetic the steps save on a 3 × 3 system
Cramer's rule for three unknowns needs four determinants. With the rule of Sarrus each one is a sum of 6 triple products, so the whole method is 24 triple products, 48 multiplications, about 20 additions and subtractions, and 3 divisions, before the check. One sign slip anywhere changes the answer, and nothing on the page says so.
Checked by hand afterwards
- Three more substitutions with fractions
- A wrong answer found only at the very end
- No way to tell which of the 48 multiplications went wrong
Compared step by step
- Each determinant printed with its full expansion
- Your Dy against the calculator's: the first mismatch is the slip
- Check table: left side, right side, holds for every equation
Exact fractions matter here too. A decimal answer such as 0.333333 substituted back never gives the right side exactly; the fraction 1/3 does, so the check column can say "holds" rather than "close enough". Decimals are still printed under the fractions, rounded to 6 places, for anyone who needs them.
Questions students ask about linear systems
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