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Old print of a lion tamer
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

\[\Large{ax+by=c}\]

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

\[\Large{y=mx + b.}\]

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.

Graph of the lines l and m.

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

\[\Large{ax+by=c}\]

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

\[\Large{y=-\frac{a}{b}x+\frac{c}{b}.}\]

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 \(\normalsize{k:\; y=mx+b}\).

Answers

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

\[{\Large y = -\frac{1}{4}x + \frac{5}{2}.}\]

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.

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This article is from the free online course:

Maths for Humans: Linear, Quadratic & Inverse Relations

UNSW Sydney