The Lesson

A linear equation is an equation that shows a straight line when it is plotted on a graph.

An Example of a Linear Equation

An example of a linear equation is shown below:

In this example, there are two variables (x and y). None of the variables have a number written to the right and above it (that is, the highest exponent is 1).

Linear Equations as Equations of Lines

The linear equation has two variables: x and y.

y equals 2 x plus 1 Pairs of values of x and y will make both sides of the equation equal. In the table below, pairs of values of x and y are chosen in the left hand column. They are substituted into the equation (y = 2x + 1) in the right hand column. Both sides of the equals sign are equal.
x = 1, y = 3 3 = 2 × 1 + 1 \(\:\:\:\) = 2 + 1 = 3
x = 2, y = 5 5 = 2 × 2 + 1 \(\:\:\:\) = 4 + 1 = 5
x = 3, y = 7 7 = 2 × 3 + 1 \(\:\:\:\) = 6 + 1 = 7
Treat the value of x as the x-coordinate and the value of y as the y-coordinate. These pair of x and y values can be plotted on a graph as Cartesian coordinates (x, y). When they are plotted, they make a line:

linear equation as a line

Forms of Linear Equations

Linear equations come in many forms.

What a Linear Equation Is...

Let's see what parts a linear equation can have:

What makes a linear equations A linear equation can contain:

...And What It Isn't

Now let's see what a linear equation cannot have. If you see any of these, it isn't a linear equation.

What does not make a linear equation The variables in the linear equation (the y and x) cannot contain:
  • Exponents - variables can only appear as x and y, not as x2 or y3
  • Roots - variables cannot appear as √x or ∛y