Number of Solutions of an Equation
An equation can have one solution, no solution or infinitely many. Learn how to tell which case you have for a linear equation, a pair of linear equations and a quadratic, with worked examples and practice.
An equation can have one solution, no solution or infinitely many solutions, and you can tell which case it is before you finish solving. A linear equation ax + b = 0 has one solution if a ≠ 0, no solution if a = 0 and b ≠ 0, and infinitely many if a = 0 and b = 0. This page explains the rule for one equation, for a pair of equations and for quadratics.
One linear equation: one solution, no solution, infinite solutions
Move everything to one side until it looks like ax = c. Then three things can happen.
| Equation | After simplifying | Solutions |
|---|---|---|
| 2x + 3 = 7 | x = 2 | one |
| 2x + 3 = 2x + 5 | 3 = 5, false | none |
| 2x + 3 = 2x + 3 | 3 = 3, always true | infinitely many |
The solution of ax + b = 0 is x = −b/a. For 2x + 3 = 7 you get 2x − 4 = 0, so x = 4/2 = 2.
A pair of linear equations: one, no or infinite solutions
Each equation ax + by = c is a line. The solutions are the points that lie on both lines.
| Pair | Lines | Solutions |
|---|---|---|
| x + y = 5 and x − y = 1 | cross | one: x = 3, y = 2 |
| x + y = 5 and x + y = 7 | parallel | none |
| x + y = 5 and 2x + 2y = 10 | the same line | infinitely many |
For a1x + b1y = c1 and a2x + b2y = c2 compare the ratios. If a1/a2 ≠ b1/b2, there is one solution. If a1/a2 = b1/b2 ≠ c1/c2, there is none. If all three ratios are equal, there are infinitely many. In the last example the ratios are 1/2, 1/2 and 5/10.
Non-zero solutions
If the right sides are 0, x = 0 and y = 0 is always a solution (the trivial one). A non-zero solution exists only when the lines coincide. x + y = 0 and 2x + 2y = 0 have infinitely many, such as (1, −1). x + y = 0 and x − y = 0 have only (0, 0).
Quadratic equations and polynomials: number of real solutions
For ax2 + bx + c = 0 the discriminant D = b2 − 4ac decides.
| Equation | D | Real solutions |
|---|---|---|
| x2 − 5x + 6 = 0 | 1 | two: 2 and 3 |
| x2 − 4x + 4 = 0 | 0 | one: 2 |
| x2 + x + 1 = 0 | −3 | none (two complex solutions) |
In general, an equation of degree n has at most n real solutions, and exactly n complex solutions if you count repeated ones twice.
Practice problems with answers
Frequently asked questions
When does an equation have infinitely many solutions?
When both sides simplify to the same expression, such as 2x + 3 = 2x + 3. Every number is then a solution.
When does an equation have no solution?
When it simplifies to a false statement, such as 3 = 5. In a pair of equations, this happens for parallel lines.
What is the solution of the equation ax + b = 0?
x = −b/a if a ≠ 0. For a = 0 there is no solution if b ≠ 0 and every number is a solution if b = 0.
Related: number systems explains the sets of numbers in which solutions can lie, including real and complex numbers.
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