Standard Form to Slope-Intercept Form Calculator

Convert linear equations from Ax + By = C to y = mx + b form.

Enter the coefficients A, B, and C from your standard form equation to get the slope-intercept form, slope (m), and y-intercept (b).

Examples

Click on an example to see how the conversion works.

Basic Conversion

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A standard example with positive integer coefficients.

A: 2

B: 3

C: 6

Negative Coefficients

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An example involving negative numbers.

A: 4

B: -2

C: 8

Zero Coefficient A

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A horizontal line where the coefficient of x is zero.

A: 0

B: 5

C: 10

Fractional Result

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An equation that results in fractional slope and y-intercept.

A: 3

B: 4

C: 7

Other Titles
Understanding Standard Form to Slope-Intercept Form Conversion
A comprehensive guide to converting linear equations, understanding their components, and applying them in various contexts.

The Two Forms of Linear Equations

  • Understanding Standard Form (Ax + By = C)
  • Understanding Slope-Intercept Form (y = mx + b)
  • Why conversion between these forms is important
Linear equations are fundamental in algebra and represent straight lines on a graph. They can be expressed in several forms, but the two most common are the standard form and the slope-intercept form.
Standard Form: Ax + By = C
The standard form of a linear equation is Ax + By = C, where A, B, and C are constants. This form is particularly useful for finding the x and y-intercepts of a line easily. By convention, A is usually a non-negative integer, and A, B, and C are integers.
Slope-Intercept Form: y = mx + b
The slope-intercept form, y = mx + b, is powerful because it directly reveals two key properties of the line: its slope (m) and its y-intercept (b). The slope indicates the steepness and direction of the line, while the y-intercept is the point where the line crosses the vertical y-axis.

Key Characteristics

  • Standard Form (2x + 3y = 6) is great for finding intercepts.
  • Slope-Intercept Form (y = -2/3x + 2) immediately tells you the slope is -2/3.

Step-by-Step Conversion Guide

  • The goal: Isolate y on one side of the equation
  • Handling coefficients and constants
  • Deriving the slope (m) and y-intercept (b)
Converting from standard form to slope-intercept form is a straightforward algebraic process. The main objective is to solve the equation for y.
The Conversion Process
1. Start with the standard form equation: Ax + By = C
2. Subtract the x term from both sides: By = -Ax + C
3. Divide all terms by the coefficient B: y = (-A/B)x + (C/B)
4. Identify the slope and y-intercept: Comparing this to y = mx + b, we can see that m = -A/B and b = C/B.

Conversion Examples

  • Given 2x + 4y = 8, we get 4y = -2x + 8, which simplifies to y = -0.5x + 2.
  • Given 5x - y = 3, we get -y = -5x + 3, which simplifies to y = 5x - 3.

Real-World Applications

  • Modeling real-world scenarios with linear equations
  • Interpreting slope as a rate of change
  • Using the y-intercept as a starting point
The slope-intercept form is incredibly useful in modeling real-world situations where there is a constant rate of change.
Example: Business and Finance
A company's profit (y) might be modeled by an equation where 'x' is the number of units sold. The slope (m) would represent the profit per unit, and the y-intercept (b) would represent the fixed costs (a negative value) or a base income.
Example: Physics
In kinematics, the position (y) of an object moving at a constant velocity can be described by a linear equation. The slope (m) is the velocity, and the y-intercept (b) is the initial position.

Practical Scenarios

  • Cost analysis: C = 10q + 500 (Cost is $10 per quantity plus $500 fixed cost).
  • Temperature conversion: F = 1.8C + 32 (Fahrenheit depends on Celsius).

Special Cases and Common Pitfalls

  • Handling horizontal and vertical lines
  • What happens when B=0?
  • Avoiding common algebraic mistakes
Horizontal Lines (A=0)
When A=0, the standard form is 0x + By = C, or simply By = C. Solving for y gives y = C/B. This is a horizontal line with a slope of 0. For example, 2y = 6 becomes y = 3.
Vertical Lines (B=0)
When B=0, the standard form is Ax = C. This equation cannot be written in slope-intercept form because you cannot solve for y. This represents a vertical line, x = C/A, which has an undefined slope. Our calculator flags this as an error because y = mx + b cannot represent a vertical line.
Common Mistakes
A common mistake is forgetting to divide the constant C by B. Remember that both the -Ax term and the C term must be divided by B.

Edge Case Examples

  • Horizontal Line: 3y = 9 --> y = 3 (slope is 0)
  • Vertical Line: 2x = 8 --> x = 4 (undefined slope)

Mathematical Derivation and Proof

  • The algebraic foundation of the conversion
  • Ensuring equivalence between the two forms
  • Relationship between coefficients and slope/intercept
The conversion from standard form to slope-intercept form is a simple but rigorous algebraic manipulation that preserves the equality of the equation.
Derivation Steps
1. Premise: We are given Ax + By = C, with the condition that B ≠ 0.
2. Isolation of the y-term: Using the subtraction property of equality, we subtract Ax from both sides: Ax - Ax + By = C - Ax, which simplifies to By = -Ax + C.
3. Solving for y: Using the division property of equality, we divide every term by B: (By)/B = (-Ax)/B + C/B.
4. Final Form: This simplifies to y = -(A/B)x + (C/B). This equation is now in the form y = mx + b.
Conclusion
Through this derivation, we have proven that for any linear equation in standard form where B is not zero, there is an equivalent slope-intercept form where the slope m = -A/B and the y-intercept b = C/B.

Formal Derivation

  • If Ax + By = C, then By = -Ax + C.
  • If By = -Ax + C and B≠0, then y = (-A/B)x + (C/B).