Linear Equations

Linear Equations are equations in which every term is either a constant or the product of a constant with a single variable that is raised to the power \(1\). They are the equations with graphs that are straight lines.

For example, these are all linear equations:

• \(y = x\)
• \(y = 3\)
• \(y = 3x + 4\)
• \(15 x = 6y + 7\)
• \(\dfrac{1}{3}\;y = 12 x - 6\)

These are not linear equations:

• \(y = x^2 + 6\)
• \(y^2 = x + 7\)
• \(y = \sqrt{x + 3}\)
• \(y = 2^x + 3\)
• \(y = \sin(x + 4)\)

If an equation includes a power of \(x\) other than \(x^1 = x\) or \(x^0 = 1\), a square root, a sine, a cosine, an exponential etc, it is not linear.

Example

Different Forms of Linear Equations

Linear equations have a number of different possible forms. All include constants (like 0 or -236), and the variables are only ever raised to the power \(1\).

Here are some different ways of writing the linear equation \(y = x + 1\):

• \(y = x + 1\) (gradient-intercept form)
• \(y - 1 = 1(x - 0)\) (point-gradient form)
• \(\dfrac{y - 1}{5 - 3} = \dfrac{x - 0}{4 - 2}\) (2 point form)
• \(x - y + 1 = 0\) (general form)

and there are many other different ways to write the same equation. Let's look at the four different forms of linear equation used above.

Gradient-Intercept Form

Point-Gradient Form

Two Point Form

Two points are enough to nail down a linear equation. If the points are \((x_1,y_1)\) and \((x_2,y_2)\), then the equation is

For example, the linear equation satisfied by both the points \((3,4)\) and \((5,6)\) is

General Form

Sometimes we are asked to put linear equations into general form. This is particularly relevant when we need to do matrix operations to manipulate systems of linear equations. We'll talk more about that in the articles on matrices.

Here's the general form of the equation of a straight line. The \(A\), \(B\) and \(C\) are constants, and \(A\) and \(B\) can't both equal zero. Note that the term in \(x\) comes first:

For example, the equation \(2x + 3y - 5 = 0\) is in general form with

• \(A = 2\)
• \(B = 3\)
• \(C = -5\)

Function Notation

It is common to see linear equations written in function notation. Some examples include:

• \(f(x) = -2x + 4\)
• \(g(q) = 3q - 2\)
• \(h(a) = 17a - 254\)

The Identity Function

Sometimes we call the linear equation \(f(x) = x\) the identify function because it doesn't change \(x\) at all. \(x\) maintains its "identity".

The graph of the identify function looks like this:

Notice that it makes an angle of \(45^\circ\) with the positive \(x\)-axis. Its gradient is equal to 1.

Constant Functions

Constant functions are also linear equations. Of course, only one of the variables can be kept constant, or you'd just get a point.

The graphs of \(f(x) = \text{constant}\) look like this:

The graphs of \(x = \text{constant}\) are vertical lines.

Some examples of constant equations are

• \(f(x) = 17\): the points satisfying this equation are \((1,17), (243,17), (-23,17), (\text{anything you like}, 17)\).
• \(x = 2\): the points satisfying this equation are \((2,-243),(2,0), (2,27), (2,\text{anything you like})\).

Conclusion

That's all I have to say about linear equations for now. You might like to have a read about the ways you can use linear equations to tell whether lines are parallel or perpendicular, or to find out how you solve systems of linear equations in some of the other articles. Have fun!