Linear Equations
(KS3, Year 7)

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).

Interactive Widget

You can use this interactive widget to create a graph of your linear equation. Use the buttons to change the values of the linear equation.
Change the Equation

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
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See Also

What are Cartesian coordinates? What is a linear equation (in general form)? What is a linear equation (in slope-intercept form)? What is a linear equation (in slope-point form)?