Forms of Linear Equations

Concept

Slope-Intercept Form

A linear equation or linear function can be written in the following form called the slope-intercept form.

$y = m x + b$

In this form, $m$ is the slope and $b$ is the $y -$ intercept. These are the general characteristics or parameters of the line. They determine the steepness and the position of the line on the coordinate plane. Consider the following graph.

The graph of the linear function y=2*x+1 with a slope of 2 (2 rise, 1 run) and a y-intercept at (0, 1)

This line has a slope of $2$ and a $y -$ intercept of $1 .$ The equation of the line can be written in slope-intercept form using these values.

y = m x + b ⇓ y = 2 x + 1

Concept

Point-Slope Form

A linear equation with slope $m$ through the point $(x_{1}, y_{1})$ is written in the point-slope form if it has the following form.

$y - y_{1} = m (x - x_{1})$

In this point-slope equation, $(x_{1}, y_{1})$ represents a specific point on the line, and $(x, y)$ represents any point also on the line. Graphically, this means that the line passes through the point $(x_{1}, y_{1}) .$

It is worth mentioning that the point-slope form can only be written for non-vertical lines.

Why

Derivation of the Formula

The point-slope form can be derived by using the Slope Formula. To do so,

(x, y)

— which represents any point on the line — is substituted for

(x_{2}, y_{2})

into the formula.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstituteII

$x_{2} = x$ , $y_{2} = y$

m = \frac{y - y _{1}}{x - x _{1}}

MultEqn

$LHS \cdot (x - x_{1}) = RHS \cdot (x - x_{1})$

m (x - x_{1}) = y - y_{1}

RearrangeEqn

Rearrange equation

y - y_{1} = m (x - x_{1})

Concept

Standard Form of a Line

In the standard form of a line all $x -$ and $y -$ terms are on one side of the linear equation or function and the constant is on the other side.

$A x + B y = C$

In this form, $A,$ $B,$ and $C$ are real numbers. It is important to know that $A$ and $B$ cannot both be $0 .$ Different combinations of $A,$ $B,$ and $C$ can represent the same line on a graph. It is preferred to use the smallest possible whole numbers for $A,$ $B,$ and $C$ and it is also better if $A$ is a positive number.

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Why