What is a line and what isn't?

With all these different types of line equations, how do I actually tell if an equation is a line or not?

This is a really good question. Usually it’s linear relationships (line equations in other words) that are the first type of relationship that you learn about when studying algebra. If you haven’t seen many other types of equations then you probably have very little to compare them to, so it can be difficult to see what it is that makes a line equation a line equation.

Let’s have a look at a few different forms of the line equation:

$\boldsymbol{\large{\text{Slope/intercept form: }\;y=-5x+8}}$

$\boldsymbol{\large{\text{Point/slope form: }\;y-3=-5(x-1)}}$

$\boldsymbol{\large{\text{General form: }\;10x+2y=16}}$

All of those equations above represent the same line. They are just written in different ways. The reason we have different forms of the line equation is simply because different forms are more useful or easier to use in different situations.

The things they all have in common and the things that make them line equations are:

All the terms are either just a number or a variable (the $\boldsymbol{x}$ and $\boldsymbol{y}$ ) multiplied by a number.
All the variables have an exponent of 1.

That definitely deserves some examples to explain what I mean. Remember that if something has an exponent of 1 we normally don’t write the exponent at all because, for example, $\boldsymbol{5^1=5}$ . So these things are all not line equations:

$\boldsymbol{\large{y=x^2+4\quad\quad y^3=x\quad\quad y=(x+1)^2\quad\quad y=\frac{1}{x}}}$

Remember that $\boldsymbol{\frac{1}{x}=x^{-1}}$ so that last one is not a line equation for the same reason as the others. The exponent of the $\boldsymbol{x}$ is not 1, it is -1.

All the variables in the following equations have an exponent of 1. However, they still are not line equations because they have terms other than just numbers or variables multiplied by numbers:

$\boldsymbol{\large{y=2^x-7\quad\quad y=\sin{x}+5x\quad\quad y=xz\quad\quad y=\log{x}}}$

Don’t worry if you don’t know what some of those things are, as long as you recognise they’re not just $\boldsymbol{x}$ multiplied by a number.

Now, there are some things that meet the criteria above that look a little different than what you may be used to thinking of as a line equation. For example, these definitely are line equations:

$\boldsymbol{\large{y=3\quad\quad y=\frac{2}{3}x-\frac{7}{11}\quad\quad 2x-7y+3z=24}}$

These line equations have either more or less terms than you’re probably used to or, in the case of the middle one, just have fiddly little numbers. Those two criteria above, though, put no restriction on how many terms there must be, nor on how cuddly the numbers should be.

That’s all for now. If you have any questions, let me know.

What is a line and what isn’t?

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