Algebra Calculator
Choose an equation type, enter its coefficients and get the solution with the working shown step by step. This calculator solves three kinds of equation only: one linear equation ax + b = cx + d, a system of two linear equations in x and y, and a quadratic ax² + bx + c = 0. It is not a computer algebra system, so it cannot factor, expand or simplify expressions.
Last updated November 4, 2026
Solves three equation types only: one linear equation, a system of two linear equations, or a quadratic. It is not a symbolic algebra system and does not parse typed equations. For quadratics only, see the quadratic formula calculator.
Result
x = 4
One solution.
| a - c | 2 |
|---|---|
| d - b | 8 |
Show the work
Formula: x = (d - b) / (a - c)
- Equation: 2x + 3 = 0x + 11.
- Move x terms to one side and numbers to the other: (a - c)x = d - b, so (2 - 0)x = 11 - 3, that is 2x = 8.
- Divide: x = 8 / 2 = 4.
- Check: with x = 4, the left side is 11 and the right side is 11.
Result: x = 4
The three formulas used
x = (d − b) ÷ (a − c)
For a single linear equation ax + b = cx + d, collect the x terms on one side and the plain numbers on the other. That gives (a − c)x = d − b, so x = (d − b) ÷ (a − c) whenever a − c is not zero. If a − c is zero the x terms cancel. Then the equation has no solution when d − b is not zero, and every number is a solution when d − b is zero as well.
For two linear equations a1x + b1y = c1 and a2x + b2y = c2, the calculator uses Cramer's rule. The determinant is D = a1·b2 − a2·b1, and when D is not zero, x = (c1·b2 − c2·b1) ÷ D and y = (a1·c2 − a2·c1) ÷ D. When D is zero the two lines are parallel or identical, so the answer is either no solution or a whole line of solutions.
For a quadratic ax² + bx + c = 0, the calculator uses the same engine as the quadratic formula calculator: x = (−b ± √(b² − 4ac)) ÷ 2a. A positive discriminant gives two real roots, zero gives one repeated root and a negative value gives a pair of complex roots.
See how we calculate for the formulas, sources and review schedule behind every calculator.
Also useful alongside this calculator: Scientific Calculator.
Exact fractions, no rounding noise
Computers store decimals such as 0.1 as binary approximations, which can break an exact zero test. Every number you type is converted to an exact fraction, so 0.1 becomes 1/10, and all the arithmetic is done on whole numbers and reduced with the greatest common divisor.
The result is that 0.1x + 0.2 = 0.3 gives exactly x = 1, and 3x = 1 gives the fraction 1/3 together with the decimal 0.33333333. Inputs may have up to 12 decimal places and must lie between −1,000,000,000 and 1,000,000,000. Quadratic roots are shown as decimals because they often involve square roots.
No solution and infinitely many solutions
These are valid answers, not errors. The equation 3x + 1 = 3x + 5 reduces to 1 = 5, which is never true, so there is no solution. The equation 2x + 4 = 2x + 4 reduces to 0 = 0, which is always true, so every x works.
The same idea applies to two equations. The pair x + y = 1 and x + y = 2 describes two parallel lines that never meet. The pair 2x + 4y = 6 and x + 2y = 3 describes the same line twice, so every point on it is a solution.
What this calculator does not do
You enter coefficients, not typed equations. It does not factor, expand or simplify polynomials, handle inequalities, or solve equations with exponents above 2, fractions of x, or three or more unknowns. For a quadratic alone, with its vertex, use the quadratic formula calculator.
How to use the Algebra Calculator
- Choose the equation type: one linear equation, two linear equations, or a quadratic.
- Rewrite your equation in the matching form, then enter each coefficient. Use a minus sign for negative values and 0 for a missing term.
- Read the solution, or the message that there is no solution or infinitely many.
- Open Show the work to follow each step with your numbers, then use Copy, Print or Share.
Worked example
Solve 2x + 3 = 11, so a = 2, b = 3, c = 0 and d = 11. Subtracting c from a gives 2 and subtracting b from d gives 8, so x = 8 ÷ 2 = 4. Check: 2 × 4 + 3 = 11. The table also lists two more cases solved with the same method.
| Equation | Method | Result |
|---|---|---|
| 2x + 3 = 11 | x = (11 − 3) ÷ (2 − 0) | x = 4 |
| 3x + 1 = 3x + 5 | a − c = 0 and d − b = 4 | No solution |
| x + y = 3 and x − y = 1 | D = −2, Dx = −4, Dy = −2 | x = 2, y = 1 |
| x + y = 1 and x + y = 2 | D = 0, constants differ | No solution |
Frequently asked questions
Can this calculator solve any algebra problem?
No. It solves one linear equation in x, a system of two linear equations in x and y, and a quadratic. It does not simplify, factor or expand expressions.
Why does it say "no solution" instead of an error?
Some equations are genuinely impossible, such as 3x + 1 = 3x + 5. That is a correct mathematical result, so the calculator reports it and explains why.
What does "infinitely many solutions" mean?
Both sides of the equation are always equal, so any value of x works. For two equations it means they describe the same line, and every point on that line is a solution.
Why is 0.1x + 0.2 = 0.3 exactly 1 here?
The calculator turns decimals into exact fractions before calculating, so it avoids the tiny floating-point errors that give answers like 0.9999999999999998 elsewhere.
How do I enter an equation like 5x = 2x + 9?
Match it to ax + b = cx + d: here a = 5, b = 0, c = 2 and d = 9, which gives x = 9 ÷ 3 = 3. Use 0 for any term that is missing.