Linear Equations

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. 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:

Forms of Linear Equations

Linear equations come in many forms.

• The general form of a linear equation is:
  
  the general form of a linear equation
• The slope-intercept form of a linear equation is: m is the slope and c is the y-intercept.
  
  the slope-intercept form of a linear equation
• The slope-point form of a linear equation is:
  
  m is the slope of the line, and the point (x₁, y₁) is a point (in Cartesian coordinates) on the line.
  
  the slope-point form of a linear equation

What a Linear Equation Is...

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

• Variables (such as y and x above)
• Constants (such as 1 above)
• Coefficients - a constant in front of a variable (such as 2 above)

...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. 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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