# Linear Equations

## What Is an Equation?

An equation shows us that two things are equal. An equals sign (=) is used to show that things are equal. What is to the left of the equals sign is equal to what is on the right of the equals sign. Here is an example of an equation:

The equation above tells us that x + 1 is equal to 2. The quantities on either side are called expressions and can use numbers (1 and 2), letters (x) and operators (+). The equation is expressing a relationship between different numbers and symbols.

## Dictionary Definition

The Oxford English Dictionary defines an equation as "a formula affirming the equivalence of two quantitative expressions, which are for this purpose connected by the sign."

## Robert Recorde and Equations

The first recorded use of the equals sign = in an equation (as well as the plus + and minus − signs in an English book) was by Robert Recorde (1510 - 1558). It appears in his book 'The Whetstone of Witte':

It is equivalent to '14x + 15 = 71' in today's notation. He chose two parallel lines of equal length, "bicause noe 2 thynges can be moare equalle."

## What Is a Linear Equation?

A linear equation is a type of equation. A linear equation has two variables (x and y). It shows a straight line when plotted on a graph. Here is an example of a linear equation:

Here is this linear equation plotted on a graph:

## Equations and Linear Equations

Equations show that two mathematical expressions are equal to each other. Linear equations are the simplest type of equation and show straight lines when plotted. In this mini-curriculum, you will learn about equations and linear equations.

Equation
An equation tells us that two things are equal. Equality is denoted by the equals sign (=).
Here is an example of an equation:

It tells us that x is equal to 2. Equations can show the relationships between combinations of letters and numbers.

Linear Equation
A linear equation is an equation that shows a straight line when it is plotted on a graph.
Here is an example of a linear equation and the line it shows when plotted on a graph:

## Lines, Slope and Y-Intercept

Linear equations can be written in different forms. The slope-intercept form of a linear equation lets us read off what the slope and y-intercept of a line are. In this mini-curriculum, you will learn what the slope and y-intercept of a line are and how you can read them off from a linear equation.

Linear Equation in Slope-Intercept Form
A linear equation in slope-intercept form is written as y = mx + c, where m is the slope and c is the y-intercept.
Here is an example of a linear equation in slope-intercept form and the line it shows when plotted on a graph:

- The slope is 2 because this is the number in front of the x.
- The y-intercept is 1 because this is the number added after the x.

Slope
The slope of a line is its steepness. The larger the slope, the steeper the line.

Find Slope from Line
To find the slope of a line, divide how far up the line goes by how far across the line goes.
Here is an example of a line with a slope of 1. It goes up 4 units and goes across 4 units, so its slope is (4 ÷ 4 = ) 1.

Find Slope from Linear Equation
To find the slope of a line from a linear equation, look at the number in front of the x.
Here is a linear equation that shows a line with a slope of 2 because there is a 2 in front of the x:

Y-Intercept
The y-intercept is the y-coordinate of the point where a line crosses the y-axis.

Find Y-Intercept from Line
To find the y-intercept of a line, read off the y-coordinate of where the line crosses the y-axis.
Here is an example of a line with a y-intercept of 1.

Find Y-Intercept from Linear Equation

To find the y-intercept of a line from a linear equation, look at the constant term (that is not in front of x or y).
Here is a linear equation that shows a line with a y-intercept of 1 because 1 is the constant term:

Find Equation of a Line
To find the equation of a line, find the slope and write it in front of the x and find the y-intercept and write it as the constant.
A line with a slope of 2 and a y-intercept of 1 will have an equation of:

y = 2x + 1

Read more about how to find the slope from a linear equation
Read more about how to find the y-intercept from a linear equation

## Slope-Point Form

Linear equations can be written in different forms. The slope-point form of a linear equation lets us read off what the slope of a line is and the Cartesian coordinates of a point the line passes through. In this mini-curriculum, you will learn about the slope-point form of a linear equation.

Slope-Point Form
A linear equation in slope-point form is written as y − y1 = m(x − x1), where m is the slope and (x1, y1) are the Cartesian coordinates of a point on the line.
Here is an example of a linear equation in slope-point form and the line it shows when plotted on a graph:

- The slope is 2 because this is the number in front of the brackets.
- The line passes through the point with Cartesian coordinates (1, 3) because of the numbers after the − signs

Find Slope from Linear Equation in Slope-Point Form
To find the slope of a line from a linear equation in slope-point form, look at the number in front of the brackets.

Find Y-Intercept from Linear Equation in Slope-Point Form
To find the y-intercept of a line from a linear equation in slope-point form, substitute x = 0 into the equation and rearrange it.

y − 3 = 2(x − 1)
y − 3 = 2(0 − 1) ⇒ y = 1

Read more about how to find the slope from a linear equation in slope-point form
Read more about how to find the y-intercept from a linear equation in slope-point form

## General Form

Linear equations can be written in different forms. The general form of a linear equation does not let us read off anything about the line it represents. In this mini-curriculum, you will learn about the general form of a linear equation.

General Form
A linear equation in general form is written as ax + by + c = 0.
Here is an example of a linear equation in general form and the line it shows when plotted on a graph:

Find Slope from Linear Equation in General Form
The slope of a line can be found from a linear equation in general form.

Find Y-Intercept from Linear Equation in General Form

The y-intercept of a line can be found from a linear equation in general form.

Convert General Form to Slope-Intercept Form
A linear equation in general form can be converted to slope-intercept form.

Read more about how to find the slope from a linear equation in general form
Read more about how to find the y-intercept from a linear equation in general form
Read more about how to convert a linear equation from general to slope-intercept form

## Equation of a Line Between Two Points

If we know the Cartesian coordinates of two points on a line, the equation of the line can be found. In this mini-curriculum, you will learn about how to find the linear equation of a line between two points.

Find Equation of Line Between Two Points
The equation of a line where (x1, y1) and (x2, y2) are the Cartesian coordinates of two points on the line is: