Linear equations

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Linear equations form a straight line when graphed on a coordinate plane. For example,

is a linear equation. The solutions to the equation fall on a straight line.

Nonlinear equations do not create straight lines when graphed on the coordinate plane. For example,

–

is a nonlinear equation. The solutions to the equation fall on a curve.

You can tell if an equation is linear by looking at it. Linear equations usually have constants and always have at least one variable. Linear equations never have a variable raised to a power other than 1. They also never have variables within absolute value signs or under radical signs. Here are a few examples of linear equations:

x
=
5
y
+
x
–
1
=
0
x
=
6
–
3
y
y
4
=
2

Here are some examples of nonlinear equations:

y
2
=
x
+
1
y
=
x
–
3
y
–
2
=
x
3
y
=
absolute value bar
2
x
+
1
absolute value bar

Try some practice problems!

Identify linear and nonlinear functions: graphs and equations

Forms of linear equations

Linear equations can be written in many different forms! Read on to learn about some of the common forms.

Slope-intercept form

Linear equations can be written in

slope-intercept form

Equations in slope-intercept form follow this structure:

In slope-intercept form,

is the slope and

is the

-intercept

The equation

–

is in slope-intercept form. It has a slope of

–

and a

-intercept

of 2. The equation is graphed below.

Point-slope form

You can also write linear equations in

point-slope form

Equations in point-slope form follow this structure:

–

(

–

)

In point-slope form,

and

form the ordered pair

(

)

which is a point on the line. The slope of the line is

The equation

–

(

)

is in point-slope form. It passes through the point

(

–

)

and has a slope of

The equation is graphed below.

General form

Linear equations can also be written in general

form.

Equations in general form follow this structure:

In general form,

and

cannot both be zero.

The equation

is in general form. In this equation,

and

The equation is graphed below.

Standard form

You can also write linear equations in

standard form

Equations in standard form follow this structure:

In standard form,

and

cannot both be zero. Usually, standard form is written following these rules:

A
,

B
,
and
C
are integers.
A
is greater than or equal to zero.
A
,

B
,
and
C
have no common factors other than 1.

The equation

is in standard form. In this equation,

and

The equation is graphed below.

Try some practice problems!

Convert a linear equation in standard form to slope-intercept form

Equations of horizontal and vertical lines

The equations of horizontal and vertical lines have their own forms.

Horizontal lines

If a line is horizontal, its equation will follow this structure:

In this form,

is a constant.

The equation

–

represents a horizontal line. The equation is graphed below.

Vertical lines

If a line is vertical, its equation will follow this structure:

In this form,

is a constant.

The equation

is a vertical line. The equation is graphed below.

Functions

Many linear equations are

functions

A function is a rule that assigns exactly one output for each input.

Functions can be written using function

notation.

You will often see function notation written as

(

)

read

where

is the name of the function and

is the input. For example, you can write the linear function

–

using function notation as

(

)

–

Functions can be written with names other than

and inputs other than

too. For example,

(

)

is function

with input

Skills

Slope-intercept form: find the slope and y-intercept Write a linear equation from a slope and y-intercept Convert a linear equation in standard form to slope-intercept form Identify linear and nonlinear functions: graphs and equations

Lessons

Slope-intercept form

Point-slope form

Standard form of linear equations

Linear functions

Try some practice problems!

Forms of linear equations

Slope-intercept form

Point-slope form

General form

Standard form

Try some practice problems!

Equations of horizontal and vertical lines

Horizontal lines

Vertical lines

Functions

Related

Skills

Lessons