Linear Equation Solver
Solves a linear equation of the form ax + b = 0 for the unknown x in one step. Ideal for students and professionals who need the root of a first-degree equation quickly.
About this calculator
A linear equation in one variable has the standard form ax + b = 0, where a and b are known constants and x is the unknown. Solving for x means isolating it on one side: subtract b from both sides to get ax = −b, then divide both sides by a, yielding x = −b / a. This solution is valid only when a ≠ 0; if a = 0 and b ≠ 0, the equation has no solution, and if both are 0 it has infinitely many solutions. Linear equations appear constantly in everyday problems — calculating break-even points, finding unknown distances, or solving simple rate problems. The formula x = −b / a is exact and always produces a single unique answer when a ≠ 0.
How to use
Say you need to solve 4x + 12 = 0, so a = 4 and b = 12. Enter these values into the calculator. It applies x = −b / a = −12 / 4 = −3. So x = −3 is the solution. You can verify: substitute back into the original equation: 4(−3) + 12 = −12 + 12 = 0 ✓. Try another: 7x − 21 = 0 means a = 7, b = −21, giving x = −(−21)/7 = 21/7 = 3.
Frequently asked questions
What happens when the coefficient a is zero in a linear equation?
If a = 0, the term ax disappears and the equation becomes 0·x + b = 0, or simply b = 0. If b is also zero, every real number is a solution (infinite solutions). If b is non-zero, the equation is a contradiction with no solution. In either case, the formula x = −b/a is undefined because division by zero is not allowed.
How is a linear equation different from a quadratic equation?
A linear equation contains the unknown x raised only to the first power (degree 1), producing a straight line when graphed, and always has at most one solution. A quadratic equation contains x², is degree 2, graphs as a parabola, and can have zero, one, or two real solutions. Linear equations are simpler and solvable by straightforward algebraic manipulation without needing the quadratic formula.
When would I use a linear equation solver in real life?
Linear equations model countless real-world situations. For instance, if a taxi charges a flat fee plus a per-mile rate and you know your total bill, you can set up ax + b = 0 to find the number of miles driven. Similarly, break-even analysis in business sets revenue equal to costs, which rearranges to a linear equation to find the break-even quantity. Any situation with a constant rate of change can be modeled linearly.