# How do you solve an algebra equation?

Solves an algebra equation in one variable and checks every solution.

- Page: https://www.acalculator.org/math/algebra-calculator
- JSON spec: https://www.acalculator.org/math/algebra-calculator.json
- Version: 382a0b249e23

## Default answer

Example with the default inputs (Equation 4(x - 3) + 12 = 15 - 5(x + 6)): Solving 4(x - 3) + 12 = 15 - 5(x + 6) gives x = -5/3.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| eq | Equation | The equation, in one letter. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| solutions | Solutions | Every real solution. |
| decimals | As decimals | Each solution as a decimal. |
| count | Number of real solutions | How many there are. |

## Method

A computer algebra system; each root is checked, and a numeric scan finds any missed.

## Assumptions

- Only real solutions; no answer for sin, cos or tan of the variable.

## Worked examples

1. eq = 4(x - 3) + 12 = 15 - 5(x + 6) gives solutions = x = -5/3, decimals = x ≈ -1.666666667, count = 1. Source: OpenStax College Algebra 2e, 2.2, Ex. 2. https://openstax.org/books/college-algebra-2e/pages/2-2-linear-equations-in-one-variable.
2. eq = 3/(x - 6) = 5/x gives solutions = x = 15, count = 1.

## FAQ

### How does an algebra calculator solve an equation?

It moves everything to one side, clears the fractions, and hands the result to a computer algebra system, which uses the same rules you learn in class: collecting like terms, factoring, the quadratic formula, and inverse operations such as taking a logarithm. The page then puts each solution back into your equation as you typed it and scans the number line for any solution the algebra missed.

### Can it solve equations with fractions?

Yes. For 3/(x − 6) = 5/x it multiplies out the denominators, which gives 3x = 5(x − 6), so x = 15. It then rejects any value that makes a denominator 0 in the original equation. In 2/(x + 1) − 1/(x − 1) = 2x/(x² − 1), the values 1 and −1 are excluded, and the only solution is −3.

### Why does it show an exact answer and a decimal?

An exact answer such as −5/3 or 3/2 + √29/2 keeps every digit. The decimal, −1.666666667, is easier to compare or to use in a later step. Both describe the same number, and the decimal is rounded to 10 significant figures.

### Does it simplify expressions too?

This page solves equations. To simplify an expression such as (x² − 9)/(x² + 4x + 3), use the simplify calculator; to factor a polynomial, use the factoring calculator. The math calculator does all of these from one box.

### Can I use a letter other than x?

Yes. Type the equation in any one lowercase letter, such as 3t − 4 = 11 or y² = 49, and the page solves for that letter. Two different letters, such as x + y = 3, give no answer: that needs a value for one of them. The letter e always means Euler’s number, 2.718…

### What does "No verified answer" mean?

Every solution is checked before it is shown. If a solution fails the check, if the scan finds a possible solution it cannot confirm, or if the work takes over 3 seconds, the page says "No verified answer" rather than show a list it could not check.

## Sources

- OpenStax, College Algebra 2e, section 2.2 Linear Equations in One Variable: https://openstax.org/books/college-algebra-2e/pages/2-2-linear-equations-in-one-variable
- OpenStax, College Algebra 2e, section 2.5 Quadratic Equations: https://openstax.org/books/college-algebra-2e/pages/2-5-quadratic-equations