Equation Solver (Linear, Systems and Quadratic)
Solve an equation or a pair of equations, with the working shown.
What this equation solver does
Pick the type of equation, enter the numbers, and the calculator gives the answer and a check. It covers the four kinds most students meet first:
- ax + b = c: a linear equation with one unknown.
- ax + b = cx + d: x appears on both sides.
- Two equations with two unknowns, x and y (a system).
- ax² + bx + c = 0: a quadratic equation, including complex roots.
Worked examples
| Equation | Solution |
|---|---|
| 3x + 5 = 20 | x = (20 − 5) ÷ 3 = 5 |
| 2x + 3 = 5x − 9 | x = (−9 − 3) ÷ (2 − 5) = 4 |
| 2x + y = 7 and x − y = 2 | x = 3, y = 1 |
| x² − 5x + 6 = 0 | x = 3 or x = 2 |
| x² + 2x + 5 = 0 | x = −1 ± 2i (complex roots) |
How to solve a linear equation by hand
- Collect the terms with x on one side and the numbers on the other, doing the same operation to both sides.
- Divide by the number in front of x.
- Check by putting the answer back into the original equation.
Systems of two equations
The calculator uses Cramer’s rule. With D = a×e − b×d, then x = (c×e − b×f) ÷ D and y = (a×f − c×d) ÷ D. If D is 0, the two lines are parallel or identical, so there is no single solution. You can also solve a system by substitution or elimination.
Quadratic equations
The solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. A positive discriminant gives two real roots, zero gives one repeated root and a negative discriminant gives a pair of complex roots. For more detail and steps, use the quadratic formula calculator.
Frequently asked questions
How do I solve an equation with x on both sides?
Move the x terms to one side and the numbers to the other, then divide. For 2x + 3 = 5x − 9, subtract 2x and add 9 to get 12 = 3x, so x = 4.
How do I solve two equations with two unknowns?
Use substitution, elimination or Cramer's rule, as this calculator does. The solution is the point where the two lines cross.
What if the calculator says no unique solution?
The two equations describe parallel lines (no solution) or the same line (infinitely many solutions).
Can it solve quadratic equations?
Yes, in the last tab, including equations with complex roots. For steps, use the quadratic formula calculator.
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