Enter an equation
Formula used
The calculator expands both equation sides, combines like terms, moves terms, clears denominators, reduces coefficients, and normalizes the leading sign.
How to use this calculator
- Enter one equation containing exactly one equals sign.
- Select automatic, linear, or quadratic processing.
- Choose variable names, order, precision, and normalization options.
- Press Convert to standard form.
- Review the result, coefficients, proof steps, verification, and graph.
- Copy, share, print, or download the displayed information.
Example conversions
| Input equation | Standard form | Type |
|---|---|---|
| y = 2x + 5 | 2x - y = -5 | Linear |
| y - 3 = -2(x + 4) | 2x + y = -5 | Linear |
| x/2 + y/3 = 5 | 3x + 2y = 30 | Linear with fractions |
| 0.5x - 1.25y = 3.75 | 2x - 5y = 15 | Linear with decimals |
| y = x² - 4x + 1 | x² - 4x - y + 1 = 0 | Quadratic |
| (x - 2)² + (y + 1)² = 25 | x² + y² - 4x + 2y - 20 = 0 | Circle |
Understanding equation standard form
Standard form places an equation into a consistent algebraic arrangement. For a line, variable terms appear together on one side. The constant normally appears alone on the opposite side.
A valid linear standard form is commonly written as Ax + By = C. Many courses prefer integer coefficients with no common factor. They may also require the first nonzero coefficient positive.
Equations often begin in slope-intercept or point-slope form. Converting them requires distributing multiplication and combining like terms. Terms then move across the equals sign using inverse operations.
Fractions can be removed by multiplying every term by their least common denominator. Decimal inputs can be treated as exact terminating fractions. A greatest common divisor then reduces unnecessarily large coefficients.
Quadratic equations use a different standard arrangement. One-variable quadratics usually appear as ax² + bx + c = 0. Two-variable quadratics can also represent circles, parabolas, ellipses, or hyperbolas.
The calculator identifies the polynomial degree after expansion. Linear mode rejects squared terms to prevent misleading output. Automatic mode selects the appropriate supported form without extra decisions.
Verification matters because different-looking equations may share identical solutions. Multiplying an entire equation by a nonzero number preserves equivalence. Sample substitutions provide another useful numerical confirmation method.
Graphing offers a visual check for linear equations and conics. Equivalent linear equations produce exactly the same plotted line. Intercepts reveal where that line crosses each coordinate axis.
Use exact mode when fractions are important in coursework. Decimal mode is useful for measurement-based equations and estimates. Precision controls presentation without changing the underlying symbolic calculation.
Always review the displayed steps before copying a result. They explain each rearrangement and coefficient normalization choice. This makes mistakes easier to detect and correct quickly.
Frequently asked questions
What is the standard form of a linear equation?
It is usually written as Ax + By = C, where A, B, and C are constants and A and B are not both zero.
Can this calculator clear fractions automatically?
Yes. It finds the least common denominator and multiplies every coefficient when the option is enabled.
Does it accept decimal coefficients?
Yes. Terminating decimals are converted into exact fractions internally before standardization.
Can it convert point-slope form?
Yes. Parentheses are expanded, like terms are combined, and the equation is rearranged automatically.
What happens when both variable terms cancel?
The calculator identifies either an identity with infinitely many solutions or a contradiction with no solution.
Are quadratic equations supported?
Yes. The calculator supports polynomial equations through degree two and displays a general quadratic or conic standard form.
Why make the leading coefficient positive?
A positive leading coefficient is a common classroom convention and makes equivalent answers easier to compare.
How does equation verification work?
The calculator tracks equivalence-preserving operations and, for lines, evaluates sample points against the final coefficients.
Can I save the result as a PDF?
Yes. The PDF button opens the browser print interface, where you can choose Save as PDF.
Recent calculation history
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