Linear Equation Calculator
By Shaheer Bin Ramzan
Reviewed November 2026 · Formula sources
Solve any equation of the form ax + b = cx + d. Enter the four numbers, and use 0 for a term that is missing.
How to solve a linear equation
The goal is to get x alone. Whatever you do to one side, you do to the other, so the equation stays balanced.
- Expand any brackets and combine like terms on each side.
- Move the x terms to one side by subtracting the smaller one from both sides.
- Move the plain numbers to the other side.
- Divide both sides by the coefficient of x.
- Check by substituting your answer back into the original equation.
Worked examples
One x term: 3x + 5 = 20
Subtract 5 from both sides to get 3x = 15, then divide by 3. So x = 5. In the calculator, enter a = 3, b = 5, c = 0 and d = 20.
x on both sides: 2x + 3 = 5x − 9
Move the x terms left and the numbers right: (2 − 5)x = −9 − 3, so −3x = −12 and x = 4. Check: 2(4) + 3 = 11 and 5(4) − 9 = 11.
A real-life case: comparing two taxis
One taxi charges 3 plus 2 per km, and another charges 1 plus 3 per km. The costs match when 2x + 3 = 3x + 1. Then (2 − 3)x = 1 − 3, so −x = −2 and x = 2. The fares are equal at 2 km.
Equations with no solution or infinitely many
Try 2x + 4 = 2x + 9. Subtracting 2x from both sides leaves 4 = 9, which is false, so there is no solution. Now try 2x + 4 = 2x + 4. Both sides are identical, so you get 4 = 4 and every value of x works. The calculator reports both cases instead of returning an error.
Common mistakes
The usual slip is a sign error when moving a term across the equals sign, so write the subtraction out in full. Another is dividing only one side, or forgetting to multiply every term when clearing a bracket. Always substitute your answer back in: if the two sides do not match, the working has an error. To see the same line as a graph, use the slope, distance and midpoint calculator, and for equations with x² use the quadratic formula calculator.
Frequently asked questions
What is a linear equation?
A linear equation has a variable raised only to the first power, such as 3x + 5 = 20. Its graph is a straight line, and it has at most one solution.
How do you solve an equation with x on both sides?
Subtract the smaller x term from both sides so x appears once, move the plain numbers to the other side, then divide by the coefficient of x.
What does “no solution” mean?
It happens when the x terms cancel and what is left is false, such as 0 = 5. The two sides can never be equal, whatever x is.
What does “infinitely many solutions” mean?
It happens when both sides are the same expression, so the x terms cancel and you are left with a true statement such as 0 = 0. Every value of x works.
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