# What Is a Linear Equation?

A linear equation is an equation that represents a straight line.

A linear equation is often written in the form:

On a graph, a linear equation looks like a straight line:

• y is how far up the line is.

• x is how far along the line is.

• m is the slope, or gradient, of the line.

• c is the y-intercept of the line, where it crosses the y-axis.
Note: This form of a linear equation is the slope-intercept form.

# How to Find the Equation of a Straight Line

Question: What is the equation of the line drawn below?

• What is the y-intercept, c?

The y-intercept, c, is where the line crosses the y-axis.

• What is the slope, m?

The slope, or gradient, m, is how steep a line is.

It is the change in height, divided by the change in horizontal distance.

To find the slope of the graph, draw a right-angled triangle under the line. Divide the change in y by the change in x:

A slope of 1 means the line goes up by 1 everytime it goes across by 1.

• What is the linear equation?

Insert the values of m and c into the standard form for a linear equation, y = mx + c:

Note: The coefficient of the x is a 1. The 1 does not have to be written.

# Other Forms of Linear Equations

The slope-intercept form is the most common form of a linear equation, but other forms exist.

The general form of a linear equation is:

The slope-point is another form of a linear equation:

m is the slope of the line, and the point (x1, y1) is a point on the line.

Algebra Lessons
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##### Note
NEGATIVE SLOPES AND Y-INTERCEPTS

A slope can be negative. In this case, m is negative in the linear equation (e.g. y = -3x + 1) and the line slopes downwards:

A y-intercept can be negative. In this case, the c is negative in the linear equation (e.g. y = 2x - 3) and the line crosses the y-axis below the x-axis:

# DIFFERENT FORMS OF LINEAR EQUATIONS

The many different forms of linear equations may be quite confusing. But they all have some things in common:

• There are 2 variables, y and x.

• There are constants, like 2 or c.

• The y and x have a power of 1.

You will not see other powers, such as 2:

These are quadratic equations, not linear equations.

Neither will you see powers of 3 (cubic equations) or higher.

You will not see any square roots (where the power is ½):

# CONSTANTS

Some lines don't have any slope - they go straight across or up:

• The lines that go straight across always have the same value of y. The equation of the line is:

where c is the y-intercept.

Below are some examples:

• The lines that go straight up always have the same value of x. The equation of the line is:

where b is the x-intercept.

Below are some examples: