2.2

## UNSW Sydney If lions can be tamed, then so too can lines

# Taming the algebraic duplicity of lines!

Lines are represented by linear equations. But there are a couple of options in how to present such linear equations, giving us two quite different ways of thinking about how the equation of a line relates to its geometrical form in the Cartesian plane.

In this lesson we will learn

• about the two different algebraic forms for a line

• the significance of the slope and the intercepts of a line.

## Two ways of writing a straight line

Every line can be written in the form

for some numbers $\normalsize{a,\;b,\;c}$. In fact we should require that $\normalsize{a}$ and $\normalsize{b}$ are not both zero. This form has the advantage of being the least restrictive.

Every non-vertical line can be written in the form

This is called the slope-intercept form of a line. This form emphasises the slope $\normalsize{ m}$, also called the gradient, along with the $\normalsize{y}$-intercept $\normalsize{b}$ of the line.

Q1 (M): What line cannot be written in slope-intercept form?

So we have seen two different ways of writing the same line. For example the line $\normalsize{l:\; 2x-3y=6}$ can also be written as $\normalsize{l:\; y=(2/3)x-2}$. The line $\normalsize{m: \; y=(-1/4)x+5/2}$ can also be written as $\normalsize{m:\; x+4y=10}$.

Q2 (E): What is the slope of the line ${\normalsize m: x+4y = 10}$?

Q3 (E): Find the $\normalsize{x}$- and $\normalsize{y}$-intercepts of the line $\normalsize{l:\; 2x-3y=6}$ pictured below. ## Manipulating general equations

It is often instructive to work with abstract equations rather than special cases. Having letters represent variable quantities, such as the coefficients of $\normalsize{x}$ and $\normalsize{y}$ in the equation of a line can lead to more insight into what is really going on.

For example if we multiply two numbers, say $\normalsize{3}$ and $\normalsize{4},$ then the result is $\normalsize{12},$ but the original two numbers are more or less invisible. However if we multiply two variables, say $\normalsize{a}$ and $\normalsize{b}$, then the result is $\normalsize{ab}$ and we can see more clearly where this came from.

So the general line

can be rearranged first as $\normalsize{by=-ax+c}$ and, if $\normalsize b \neq 0$, then as

Going the other way, $\normalsize{y=mx+b}$ could be rearranged as $\normalsize{mx-y=-b}$.

Q4 (M): Find the slope and $\normalsize{x}$- and $\normalsize{y}$- intercepts of $\normalsize{l:\; ax+by=c}$.

Q5 (M): Find the $\normalsize{x}$- and $\normalsize{y}$- intercepts of $\normalsize{k:\; y=mx+b}$.

A1. The line with equation $\normalsize{x=k}$ cannot be written in the $\normalsize{ y=mx+b}$ form.

A2. The line can also be written as

We can read off the slope from this form of the line: ${\normalsize m=-1/4}$.

A3. To find the $x$-intercept of the line $\normalsize{l:\; 2x-3y=6}$ we set $\normalsize{y=0}$. This gives the equation $\normalsize{l:\; 2x=6}$ so that $\normalsize{x=3}$. To find the $\normalsize{y}$-intercept we set $\normalsize{x=0}$, which gives the equation $\normalsize{l:\; -3y=6}$ so that $\normalsize{y=-2}$. Have a look at the picture of the line to convince yourself that these values are correct.

A4. The slope of this line is $\normalsize m=-a/b$. The $\normalsize{x}$-intercept of $\normalsize{l:\; ax+by=c}$ is what you get when you set $\normalsize{y=0}$ and solve for $\normalsize{x}$, namely $\normalsize{x=c/a}$. The $\normalsize{y}$-intercept is what you get when you set $\normalsize{x=0}$ and solve for $\normalsize{y}$, namely $\normalsize{y=c/b}$.

A5. The $\normalsize{x}$-intercept of $\normalsize{k:\; y=mx+b}$ is $\normalsize{-b/m}$ and the $\normalsize{y}$-intercept is $\normalsize{b}$. Note that the $\normalsize{y}$-intercept is the easy one to find.