Year 11 General Linear Equations And Their Graphs

Determining The Slope Of A Straight Line

Q: What is the slope of a line?

The slope (or gradient) of a straight line measures how steep it is — how much y changes for every 1 unit increase in x. It can be positive, negative, zero, or undefined.

Q: What is the slope formula?

There are two equivalent forms. The first is m equals rise over run, where rise is the vertical change and run is the horizontal change. The second is m equals (y2 minus y1) over (x2 minus x1), where (x1, y1) and (x2, y2) are any two points on the line.

Q: What does it mean if the slope is positive, negative, zero, or undefined?

A positive slope means the line rises from left to right. A negative slope means it falls from left to right. A slope of zero means the line is horizontal (like y equals 3). An undefined slope means the line is vertical (like x equals 2), because dividing by zero is not allowed.

Q: Why is it rise over run, not run over rise?

Because slope measures how much the height changes for each unit of horizontal travel. Putting run on top gives the reciprocal, which is the wrong measure — it would tell you horizontal change per unit of vertical change.

Q: What does slope mean in real-world problems?

In applied problems, slope is a rate of change: dollars per hour, kilometres per hour, cost per gigabyte, height per metre. The units come straight from the axes: units of y divided by units of x.

Q: Does it matter which point I pick as (x1, y1)?

No. As long as you are consistent (use the same point for both x1 and y1, and the other point for both x2 and y2), you will get the same slope. The order of subtraction in both top and bottom must be the same.

13 practice questions 2 video lessons Theory + worked examples

Theory

The slope (or gradient) of a straight line measures how steep it is — how much $y$ changes for every $1$ unit increase in $x$ . This page covers the rise-over-run formula, the two-point formula, positive/negative/zero/undefined slopes, and real-world rate-of-change problems.

The slope (or gradient) of a straight line measures how steep it is — how much $y$ changes for every $1$ unit increase in $x$ . It can be positive, negative, zero, or undefined.

There are two equivalent formulas. Rise over run reads the vertical and horizontal change directly from a graph. The two-point formula uses any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ on the line. Use whichever fits the information you are given.

In real-world problems, slope is a rate of change: dollars per hour, kilometres per hour, cost per gigabyte, height per metre. The units come directly from the axes: units of $y$ divided by units of $x$ .

Slope is rise over run — the vertical change divided by the horizontal change.

The four kinds of slope: positive, negative, zero, and undefined.

The two equivalent slope formulas:

m = \frac{rise}{run}

m = \frac{rise}{run}

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

where rise is the vertical change (top $y$ minus bottom $y$ ) and run is the horizontal change (right $x$ minus left $x$ ). $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are any two points on the line.

What the sign of $m$ tells you:

Slope	The line...
$m > 0$	rises from left to right
$m < 0$	falls from left to right
$m = 0$	is horizontal (e.g. $y = 3$ )
$m$ undefined	is vertical (e.g. $x = 2$ ) — division by zero

How to find the slope of a line

Identify two points on the line — either from coordinates given in the question, or by reading off the graph.
Substitute into $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ — keep the same order on top and bottom.
Simplify the fraction. Note the sign carefully — a slope of $- \frac{3}{2}$ means the line falls.

Rate of change. In an applied problem like $(km, cost)$ , the slope is the cost per km. The units of slope are always the $y$ -axis units divided by the $x$ -axis units.

Example 1 — Two Points, Integer Slope

Find the slope of the line through

A (2, 1)

and

B (6, 9)

Solution

$m$	$=$	$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
$m$	$=$	$\frac{9 - 1}{6 - 2}$
$m$	$=$	$\frac{8}{4} = 2$

m = 2

Example 2 — Negative Fractional Slope

Find the slope of the line through

P (- 3, 4)

and

Q (1, - 2)

Solution

Be careful with the negative $x_{1} = - 3$ : subtracting it becomes adding.

$m$	$=$	$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
$m$	$=$	$\frac{- 2 - 4}{1 - (- 3)}$
$m$	$=$	$\frac{- 6}{4}$
$m$	$=$	$- \frac{3}{2}$

m = - \frac{3}{2}

Example 3 — Horizontal Line (Zero Slope)

Find the slope of the line through

(- 1, 5)

and

(4, 5)

Solution

$m$	$=$	$\frac{5 - 5}{4 - (- 1)}$
$m$	$=$	$\frac{0}{5} = 0$

The line is horizontal.

m = 0

Example 4 — Rate of Change

A taxi charges

$ 8

after

2

km and

$ 23

after

7

km. Find the cost per km.

Solution

Treat $(km, cost)$ as points: $(2, 8)$ and $(7, 23)$ .

$m$	$=$	$\frac{23 - 8}{7 - 2}$
$m$	$=$	$\frac{15}{5}$
$m$	$=$	$3$

The cost is $$ 3$ per km.

m = 3

Common pitfalls

Putting $x$ on top. The formula is

\frac{y_{2} - y_{1}}{x_{2} - x_{1}}

, not

\frac{x_{2} - x_{1}}{y_{2} - y_{1}}

. Putting

x

on top gives the reciprocal of the right answer.

Subtracting in different orders. Use the same order top and bottom. If you use

y_{2} - y_{1}

on top, use

x_{2} - x_{1}

on the bottom — not

x_{1} - x_{2}

. A flipped sign turns a positive slope into a negative one.

Mishandling negatives. Subtracting a negative becomes adding:

3 - (- 2) = 3 + 2 = 5

. When coordinates are negative, wrap them in brackets before subtracting.

Confusing zero and undefined. A horizontal line has slope

0

(no rise per unit run). A vertical line has slope undefined (zero run, so division by zero). They are not the same.

Frequently asked questions

What is the slope of a line?

The slope (or gradient) of a straight line measures how steep it is — how much y changes for every 1 unit increase in x. It can be positive, negative, zero, or undefined.

What is the slope formula?

There are two equivalent forms. The first is m equals rise over run, where rise is the vertical change and run is the horizontal change. The second is m equals (y2 minus y1) over (x2 minus x1), where (x1, y1) and (x2, y2) are any two points on the line.

What does it mean if the slope is positive, negative, zero, or undefined?

A positive slope means the line rises from left to right. A negative slope means it falls from left to right. A slope of zero means the line is horizontal (like y equals 3). An undefined slope means the line is vertical (like x equals 2), because dividing by zero is not allowed.

Why is it rise over run, not run over rise?

Because slope measures how much the height changes for each unit of horizontal travel. Putting run on top gives the reciprocal, which is the wrong measure — it would tell you horizontal change per unit of vertical change.

What does slope mean in real-world problems?

In applied problems, slope is a rate of change: dollars per hour, kilometres per hour, cost per gigabyte, height per metre. The units come straight from the axes: units of y divided by units of x.

Does it matter which point I pick as (x1, y1)?

No. As long as you are consistent (use the same point for both x1 and y1, and the other point for both x2 and y2), you will get the same slope. The order of subtraction in both top and bottom must be the same.