Linear Equation Solver

Getting an answer takes three quick steps. The tool handles one equation or a system of two, and the fields change to match your choice.

1. Pick what to solve in the first field. Choose one equation for ax + b = 0, or two equations for a system with x and y.
2. Enter the numbers. For one equation, enter a and b. For a system, enter the six coefficients a1, b1, c1, a2, b2 and c2.
3. Press Calculate, or press Enter. The answer appears first and large, with a check that plugs it back into the equation. Press Reset to clear the form and start over.

Watch the signs. In 2x - 10 = 0, b is -10, not 10. Entering the wrong sign is the most common way to get a wrong answer from a correct method.

A linear equation in one variable has the form ax + b = 0, where a and b are numbers and x is the unknown. The solution is x = -b / a, as long as a is not 0. You get it by moving b to the other side and then dividing by a.

The two steps are simple to follow. First subtract b from both sides, which gives ax = -b. Then divide both sides by a, which gives x = -b / a. For 2x - 10 = 0, that is x = 10 / 2 = 5. Plugging 5 back in gives 2 x 5 - 10 = 0, which confirms the answer.

The formula works for any numbers, including fractions and decimals. If the answer comes out as a fraction, keep it exact when you need it exact. Our fraction calculator helps when the arithmetic stays in fractions and you want to avoid rounding.

Solve 2x - 10 = 0. Here is exactly what the calculator does.

1. Move b to the other side: 2x = 10.
2. Divide both sides by 2: x = 10 / 2 = 5.
3. Check: 2 x 5 - 10 = 10 - 10 = 0, so x = 5 is correct.

As a second check, 3x + 9 = 0 gives x = -9 / 3 = -3, and 3 x (-3) + 9 = 0. The same two steps work for any a and b, as long as a is not 0.

A system has two equations and two unknowns, written a1x + b1y = c1 and a2x + b2y = c2. The solution is the one pair of x and y that makes both equations true at the same time, which is the point where the two lines cross on a graph.

This calculator uses Cramer's rule. First find the determinant D = a1 x b2 - a2 x b1. Then find Dx = c1 x b2 - c2 x b1 and Dy = a1 x c2 - a2 x c1. The answers are x = Dx / D and y = Dy / D. For the system 2x + y = 5 and 3x - 2y = 4, D = 2 x (-2) - 3 x 1 = -7, Dx = 5 x (-2) - 4 x 1 = -14, and Dy = 2 x 4 - 3 x 5 = -7, so x = -14 / -7 = 2 and y = -7 / -7 = 1.

Always check both equations. Here 2 x 2 + 1 = 5 and 3 x 2 - 2 x 1 = 4, so the pair works. If only one equation checks out, a sign slipped in somewhere.

Sometimes there is no single answer, and the calculator tells you so instead of giving a wrong number. For one equation, that happens when a = 0. If b is also 0, the equation reads 0 = 0, which is true for every value of x, so there are infinitely many solutions. If b is not 0, the equation reads something like 5 = 0, which is never true, so there is no solution at all.

For a system, the same idea shows up in the determinant. When D = 0, the two lines are parallel or the same line. If they are the same line, every point on it is a solution, so there are infinitely many. If they are parallel but different, they never meet, so there is no solution. The system x + y = 2 and 2x + 2y = 5 has D = 0 and no solution, while x + y = 2 and 2x + 2y = 4 has D = 0 and infinitely many solutions, because the second equation is just the first one doubled.

Our decimal to fraction calculator is useful when a system gives decimal answers that you want to read as exact fractions instead.

Forgetting to flip the sign when moving b across is the most common error. In 2x - 10 = 0, moving -10 to the right gives 2x = 10, not 2x = -10. Think of it as adding 10 to both sides, which keeps the balance.

Dividing only one term by a is next. In 2x + 6 = 0, dividing by 2 means every term changes, so x + 3 = 0 and x = -3. Dividing only the 2x and leaving the 6 alone gives a wrong answer.

In a system, the most common error is mixing up which coefficient goes where in the determinant. D uses a1 with b2 and a2 with b1, and the minus sign between them matters. Write the two rows out as a small grid first, then read the determinant off the grid, and the signs take care of themselves.