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Equation: 2x + 6 = 0
Worked check: 2x + 6 = 0 → x = −6 / 2 = −3. System x + y = 10 and x − y = 2 → adding the equations gives 2x = 12, so x = 6 and y = 4. If a = 0 the equation is not a linear equation in x, so the tool says so instead of dividing by zero.
Try an example: 2x + 6 = 0 → x = −3
A linear equation is one of the simplest and most useful pieces of mathematics you will ever meet, and it turns up everywhere: working out how many hours you need to work to reach a savings target, finding the price that makes a budget balance, or checking when two phone plans cost exactly the same. The Linear Equation Solver from FistHost takes the algebra you learned at school and does it for you in seconds, with every step shown so you can follow the working and learn from it, not just copy an answer. You type in the numbers, and the solver gives you the value of x, the unknown you are trying to find.
The idea behind a linear equation is simple. An equation is a statement that two things are equal, and a linear equation is one where the unknown appears only to the power of one, never squared or multiplied by itself. The most common form is ax + b = 0, where a and b are numbers you know and x is the number you need to find. Solving it means finding the one value of x that makes the two sides equal. Once you know that value, you can check it yourself by putting it back into the original equation and seeing that both sides match.
The solver handles the two situations you meet most often. In single-equation mode, you enter the values of a and b for an equation of the form ax + b = 0, and the solver rearranges it step by step. The rule for solving ax + b = 0 is x = -b/a: you move b to the other side of the equals sign, which changes its sign, and then divide by a. The tool shows each of these steps in plain language, so you can see the equation change from 2x + 6 = 0 to 2x = -6 and then to x = -3, instead of jumping straight to the answer.
In two-variable system mode, you enter two equations with two unknowns, x and y, and the solver finds the pair of values that satisfies both equations at the same time. A single equation with two unknowns has many possible answers, but a second equation narrows it down to exactly one pair, provided the two equations are not saying the same thing in different words. The solver uses the elimination method: it adds or subtracts the two equations so that one unknown cancels out, solves for the other, and then works backwards to find the first. Every step is shown, so you can follow the elimination and check it by hand.
Answers are given as fractions as well as decimals, because a fraction is often the exact answer while a decimal is only an approximation. If the answer is x = 1/3, the solver shows you 1/3 and also 0.333..., so you can use whichever form your homework, exam or project needs. Decimals are rounded sensibly and labelled, so you always know when a number has been rounded and when it is exact.
Here is the worked check the solver is built around, so you can follow the maths yourself. Take the equation 2x + 6 = 0. The first step is to move the 6 to the other side, which changes its sign: 2x = -6. The second step is to divide both sides by 2, the number in front of x: x = -6/2. The third step is to simplify: x = -3. You can check the answer by putting -3 back into the original equation: 2 times -3 is -6, and -6 + 6 = 0, which is exactly what the equation says. The check confirms the answer.
Now try the two-variable system from the worked check: x + y = 10 and x - y = 2. Add the two equations together and the y terms cancel, because +y and -y add to zero: (x + y) + (x - y) = 10 + 2, which gives 2x = 12. Divide by 2 and you get x = 6. Put x = 6 into the first equation, 6 + y = 10, and you get y = 4. Check the second equation: 6 - 4 = 2, which matches. So the system solves to x = 6 and y = 4, and both equations are satisfied by the same pair of numbers. This is the elimination method in action, and it works for any system where the two equations are genuinely different.
Your result starts with the value of x, the headline answer, shown as a fraction and as a decimal. If the answer is a whole number, it is shown as one. If it is a fraction, the solver shows the fraction in its simplest form, so 4/6 becomes 2/3, and the decimal beside it so you can use it in a calculator or a spreadsheet. The worked steps sit below the answer, one line at a time, so you can copy the method into your notes or your homework.
Here is the honest label. When a = 0, the equation ax + b = 0 is not a linear equation in x at all, because there is no x term left to solve for. If b is also zero, the equation is true for every value of x, and if b is not zero, it is true for no value of x. Dividing by zero to force an answer would be meaningless, so the tool says so instead of dividing by zero and showing you a nonsense result. This is the correct mathematical behaviour, and it is also a useful lesson: a solver that admits it has no answer is more trustworthy than one that invents one.
The solver also stays inside its own job. It solves linear equations and two-variable linear systems, and it does not pretend to solve quadratic equations, where x is squared, or equations with powers, roots and trigonometry. Those need different methods, and using the wrong tool for them gives the wrong answer. For equations with a squared term, you need a quadratic solver, and the link to one is in the related tools below.
Linear equations are one piece of a bigger maths toolkit, and the same free calculators can handle the pieces around them. If your equation has an x squared term, the quadratic equation solver finds its roots with the same step-by-step working. If your answer comes out as a fraction that needs simplifying, or you need the largest number that divides two numbers, the GCD and LCM calculator works it out. And if the numbers in your equation come from a percentage problem, such as a discount or a price rise, the percentage calculator handles the percentage part before you set up the equation.
FistHost believes useful tools should be free, fast and honest, with no sign-up and no download standing between you and the answer. The Linear Equation Solver runs entirely in your browser, your numbers are never sent to a server, and the result is yours to keep or share however you like. If this solver helped you with your maths, the same practical thinking applies to getting your ideas online: FistHost shared hosting starts from $0.99/month, and an annual plan includes a free domain, so you can launch a blog, a portfolio or a small business website for less than the cost of a coffee each month.
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What is the answer to 2x + 6 = 0? The answer is x = -3. Move 6 to the other side to get 2x = -6, divide by 2, and check by putting -3 back in: 2 times -3 is -6, and -6 + 6 = 0.
What happens if a = 0? Then ax + b = 0 is not a linear equation in x, because there is no x term to solve for. The tool says so instead of dividing by zero, which would give a meaningless answer.
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