Algebra • Linear Equations

Linear Equation Standard Form Converter

Convert any linear equation from y = mx + b slope-intercept form into Ax + By = C standard form with automatic fraction clearing, GCD normalization, and step-by-step derivation.

Reviewed by Sanjay Samanta & Academic Review Board

Last Updated: September 2026

Verified Accurate: Linear Algebra & Coordinate Geometry

Slope-Intercept Archetype Presets

Click preset to load slope & intercept

Slope-Intercept Parameters

Enter slope (m) and y-intercept (b) for y = mx + b

Input Equation

y = -2/3x + 4

Slope (m = Rise / Run) Rate

Accepts fractions (-2/3), decimals, or integers

Y-Intercept (b) Vertical shift

Value of y when x = 0

Quick slopes:

Canonical Standard Form Result

Ax + By = C (Integer Form)

2x + 3y = 12

Coefficients: A = 2, B = 3, C = 12

Integers ✓ A,B,C ∈ ℤ

Sign Rule ✓ A ≥ 0

Coprime ✓ gcd = 1

Coefficient A

x-multiplier

Coefficient B

y-multiplier

Constant C

Constant term

X-Intercept

(6, 0)

C / A

Evaluate Y at Custom X

x = → Calculated (x, y): (3, 2.00)

Cartesian Coordinate Plane Interactive

x: 0.00, y: 0.00

Ax + By = C

Rise / Run Triangle

Y-Intercept (0, b)

X-Intercept (x₀, 0)

Drag to pan • Scroll to zoom

Step-by-Step Algebraic Derivation for y = -2/3x + 4

Slope-Intercept → Standard Form

1 Identify Starting Slope and Y-Intercept

Begin with the slope-intercept equation y = -2/3x + 4, where slope m = -2/3 and y-intercept b = 4.

2 Clear Fractions by Multiplying by Least Common Denominator (LCD)

The denominator is 3. Multiply all terms on both sides of the equation by 3:
3 × y = 3 × (-2/3x) + 3 × 4 &implies; 3y = -2x + 12

3 Move Variables (x and y) to the Left Side

Add 2x to both sides to place all variable terms on the left:
2x + 3y = 12

4 Enforce Standard Form Conventions (A ≥ 0 and Coprime Reduction)

Leading coefficient A = 2 is already positive (A ≥ 0). The greatest common divisor gcd(2, 3, 12) = 1, confirming terms are in simplest coprime integer form: 2x + 3y = 12.

Canonical Standard Form Mathematical Constraints Strict Conventions

1. Integers Only (A, B, C ∈ ℤ)

Standard form prohibits fractional or decimal coefficients. If fractions exist, multiply every term by the least common multiple of all denominators.

2. Positive Leading Term (A ≥ 0)

By algebraic convention, the coefficient of x must be non-negative. If rearranging yields -Ax, multiply every term on both sides by -1.

3. Greatest Common Divisor = 1

Coefficients must be in lowest terms: gcd(A, |B|, |C|) = 1. For example, 4x + 6y = 10 must be simplified to 2x + 3y = 5.

Direct Answer & Overview

Verified Educational Guide

How to Convert Slope-Intercept Form to Standard Form

To convert y = mx + b to standard form Ax + By = C: subtract mx from both sides to get -mx + y = b, then multiply by -1 so A is positive. If m is a fraction, multiply all terms by the denominator to clear fractions, then divide by the GCD to simplify.

Primary Mathematical Formula Slope-Intercept to Standard Form Conversion

Standard Equation

ƒ(x)

Q.E.D.

y = mx + b \implies -mx + y = b \implies Ax + By = C

Constraints: A, B, C integers; A ≥ 0; GCD(A, B, C) = 1.

Exact Formula

Input Parameters

Required

Slope (m): The coefficient of x in y = mx + b.

Y-intercept (b): The constant term in y = mx + b.

Expected Outputs

Calculated

Coefficient A: Integer coefficient of x (A ≥ 0).

Coefficient B: Integer coefficient of y.

Constant C: Integer constant on the right side.

Standard Form Equation: Ax + By = C with normalized coefficients.

Worked Numerical Example

Instant Verification

Convert y = (2/3)x - 4 to standard form

→ Subtract (2/3)x: -(2/3)x + y = -4. Multiply by -3: 2x - 3y = 12

2x - 3y = 12 (A = 2, B = -3, C = 12)

What Is Standard Form?

The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, A ≥ 0, and the greatest common divisor of |A|, |B|, and |C| is 1. This form is the most formal algebraic representation of a line.

Standard form is especially useful when working with systems of linear equations because the elimination method relies on aligned integer coefficients. It also makes it trivial to find both intercepts: the x-intercept is (C/A, 0) and the y-intercept is (0, C/B).

Ax + By = C where A, B, C ∈ ℤ, A ≥ 0, gcd(|A|, |B|, |C|) = 1

Step-by-Step Conversion Process

Move the mx term to the left side

y = mx + b → −mx + y = b

Clear any fractions or decimals

Multiply all terms by the LCD to obtain integer coefficients.

Ensure A ≥ 0

If A is negative, multiply the entire equation by −1.

Simplify by GCD

Divide A, B, C by their greatest common divisor so the coefficients are in lowest terms.

To convert back from standard form, use the Standard Form to Slope-Intercept Converter. You can also use the general form converter for more complex equation transformations.

Clearing Fractions and Normalizing Coefficients

The most challenging part of converting to standard form is handling fractional slopes. The key technique is to multiply every term by the least common denominator (LCD) to eliminate all fractions simultaneously.

Detailed Example: y = (3/5)x + (2/7)

Step 1: −(3/5)x + y = 2/7

Step 2: LCD of 5 and 7 is 35. Multiply all terms by 35:

−21x + 35y = 10

Step 3: Multiply by −1 so A ≥ 0:

21x − 35y = −10

Step 4: GCD(21, 35, 10) = 1, so coefficients are already simplified.

Step-by-Step Worked Examples

Example: Convert y = 3x − 7

Step 1: −3x + y = −7

Step 2: Multiply by −1: 3x − y = 7

Result: 3x − y = 7

Integer coefficients, A = 3 ≥ 0, GCD(3, 1, 7) = 1. Valid standard form.

Example: Convert y = −(4/3)x + 2

Step 1: (4/3)x + y = 2

Step 2: Multiply by 3: 4x + 3y = 6

Result: 4x + 3y = 6

A = 4 ≥ 0, GCD(4, 3, 6) = 1. Verify by converting back with the slope-intercept converter.

Example: Convert y = 5 (horizontal line)

Slope m = 0, so: 0x + y = 5

Result: y = 5 (or equivalently 0x + 1y = 5)

Horizontal lines have A = 0. The equation simplifies to just y = 5.

When to Use Standard Form

Systems of Equations

The elimination method requires aligned Ax + By = C form to add or subtract equations and eliminate variables.

Finding Both Intercepts

Set x = 0 to find the y-intercept (0, C/B) and set y = 0 to find the x-intercept (C/A, 0).

Linear Programming

Constraint inequalities in optimization problems are written in standard form to define feasible regions and find optimal solutions.

Matrix Representations

Standard form maps directly to matrix equations where coefficients fill the coefficient matrix.

Common Conversion Pitfalls

Leaving Fractional Coefficients

Standard form requires integers. If your slope is a fraction like 2/3, you must multiply through by the denominator before the equation qualifies as standard form.

Negative A Coefficient

By convention, A must be non-negative. If you get −2x + y = 5, multiply the entire equation by −1 to get 2x − y = −5.

Not Reducing by GCD

After clearing fractions, check if all three coefficients share a common factor. For example, 6x + 4y = 10 should be simplified to 3x + 2y = 5.

Fact-Checked & Verified • Computational Accuracy Standards

Updated September 2026 • Editorial Policy

Authored By

Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By

Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

Found an error or have an improvement suggestion? Report a calculation issue

Frequently Asked Questions

How do you convert slope-intercept form to standard form?

To convert y = mx + b to Ax + By = C: (1) Subtract mx from both sides: -mx + y = b. (2) Multiply through by -1 to make A positive: mx - y = -b. (3) If m is a fraction, multiply all terms by the denominator to clear fractions. The result should have integer coefficients with A ≥ 0.

What are the requirements for standard form?

Standard form Ax + By = C requires: (1) A, B, and C are integers. (2) A ≥ 0 (if A = 0, then B > 0). (3) A, B, and C share no common factor greater than 1 (the GCD is 1). These conventions ensure a unique standard representation.

How do you clear fractions when converting to standard form?

If the slope is a fraction like m = 2/3, the equation -2/3 x + y = b has fractional coefficients. Multiply every term by the denominator (3) to get integer coefficients: -2x + 3y = 3b. Then ensure A ≥ 0.

What is the advantage of standard form over slope-intercept form?

Standard form makes it easy to find both intercepts (set x = 0 for y-intercept, set y = 0 for x-intercept), work with integer arithmetic, and apply the elimination method for solving systems of equations.

Can a horizontal line be written in standard form?

Yes. A horizontal line y = b has slope m = 0. In standard form it becomes 0x + 1y = b, which simplifies to y = b. This is already valid standard form with A = 0 and B = 1.

How do you verify the conversion is correct?

Plug a test point (like x = 0 or x = 1) into both the original slope-intercept equation and the standard form equation. Both should produce the same y-value for the same x. You can use our Point Checker tool to verify.