About the Linear Equation Solver
This linear equation solver finds the value of the unknown in any one-variable linear equation, such as 3(x − 2) + 4 = 2x + 7 or x/2 + 1/3 = 5/6. It expands brackets, clears fractions by multiplying through by the lowest common denominator, collects the variable terms on one side and the numbers on the other, and divides — listing every step the way a teacher would write it.
The answer is exact: fractions stay as fractions (x = 13/3, not 4.333…), with the decimal value alongside. The calculator also substitutes the answer back into both sides of the original equation as a check, and recognises the special cases of an identity (infinitely many solutions) and a contradiction (no solution).
Type the equation using any single letter as the variable, with * or nothing for multiplication (2x, 3(x + 1)), / for division and brackets as needed. The equation must be linear once simplified — if a squared term survives, use the quadratic solver instead.
How to use the linear equation solver
- 1Type the equation with one "=" sign, e.g. 2(x + 3) = 5x − 9.
- 2Use any single letter as the variable and / for fractions.
- 3Read the exact solution and its decimal value.
- 4Follow the steps table to see how each term was moved.
- 5Use the substitution check to confirm both sides are equal.
Formula and method
Every linear equation in one variable can be simplified to the form ax + b = cx + d. Doing the same operation to both sides keeps the equation balanced, so you can subtract cx and b from both sides to get (a − c)x = d − b, then divide by a − c. If the equation contains fractions, multiplying both sides by the lowest common denominator first gives whole-number coefficients and makes the arithmetic easier.
If a − c = 0, the variable disappears: when d − b is also 0 the equation is an identity, true for every value; otherwise it is a contradiction with no solution. The calculator uses exact rational arithmetic throughout, so the result is the exact fraction rather than a rounded decimal.
- a, c
- Coefficients of the variable on each side
- b, d
- Constant terms on each side
- x
- The unknown being solved for
Worked examples
Brackets on one side
Expanding gives 3x − 6 + 4 = 3x − 2. Subtract 2x from both sides (x − 2 = 7), then add 2: x = 9. Check: 3(9 − 2) + 4 = 25 and 2·9 + 7 = 25.
Equation with fractions
Multiply every term by the LCD 6: 3x + 2 = 5. Subtract 2 to get 3x = 3, then divide by 3: x = 1.
Answer is a fraction
Expand: 5y + 5 = 2y − 8. Subtract 2y: 3y + 5 = −8. Subtract 5: 3y = −13. Divide by 3: y = −13/3 ≈ −4.333.
No solution
Expanding gives 2x + 6 = 2x + 1. Subtracting 2x leaves 6 = 1, which is false, so no value of x works.
Frequently asked questions
How do you solve a linear equation?+
Simplify both sides, move all variable terms to one side and all numbers to the other by adding or subtracting the same thing on both sides, then divide by the coefficient of the variable. Always do the same operation to both sides.
How do you solve linear equations with fractions?+
Multiply every term on both sides by the lowest common denominator of all the fractions. That clears the fractions and leaves an equation with whole numbers, which you solve in the usual way.
What does it mean when an equation has no solution?+
If the variable cancels out and you are left with a false statement such as 6 = 1, no value can make the two sides equal. The lines y = left side and y = right side are parallel and never meet.
What does infinitely many solutions mean?+
If the variable cancels and you get a true statement like 4 = 4, the two sides are the same expression, so every number is a solution. Such an equation is called an identity.
How can I check my answer?+
Substitute the value back into the original equation and evaluate both sides separately. If they give the same number, the solution is correct. The calculator does this check for you.