Year 11 General Linear Equations And Their Graphs

Drawing Straight-Line Graphs

13 practice questions 2 video lessons Theory + worked examples

Theory

A linear equation in two variables — like $y = 2 x + 3$ — graphs to a straight line. This page covers the three common methods (table of values, intercept method, slope-intercept form), horizontal and vertical lines, and how to rearrange equations into a graphable form.

A linear equation in two variables — like $y = 2 x + 3$ — has infinitely many solutions, each a pair $(x, y)$ . Plotting all those pairs gives a straight line.

There are three common ways to draw a line. Method 1 uses a table of values: pick two or three values of $x$ , compute the matching $y$ , plot the points and rule a line through them. Method 2 uses the intercept method: find where the line crosses each axis (set $y = 0$ for the $x$ -intercept; set $x = 0$ for the $y$ -intercept), plot the two intercepts, and rule a line through them. Method 3 uses slope-intercept form $y = m x + c$ — plot the $y$ -intercept at $(0, c)$ and step using the slope to find a second point (covered in detail in the next subtopic).

Two special cases: $y = k$ is a horizontal line through every point with $y$ -coordinate $k$ (slope $0$ ); $x = k$ is a vertical line through every point with $x$ -coordinate $k$ (slope undefined).

The line

y = 2 - x

with its two intercepts marked.

A horizontal line

y = 3

and a vertical line

x = - 2

To find the intercepts of a line:

x -intercept: set y = 0 and solve for x .

y = 0

y -intercept: set x = 0 and solve for y .

x = 0

Special lines:

Equation	Shape	Slope
$y = k$	horizontal through $(0, k)$	$0$
$x = k$	vertical through $(k, 0)$	undefined

Rearranging into $y = m x + c$ . If the equation is given as $a x + b y + c = 0$ (e.g.\ $3 x + 2 y - 12 = 0$ ), solve for $y$ . From there the $y$ -intercept and slope are easy to read off.

How to draw any straight line

Pick a method. Intercept method is fastest for non-vertical lines given in $a x + b y = c$ form. Use a table of values when the equation is already in $y = m x + c$ form. Use slope-intercept stepping when you know $m$ and $c$ directly.
Find at least two points on the line. Three is safer — a third point checks your arithmetic.
Plot the points and rule a straight line through them. Extend the line slightly beyond the plotted points, and label it.

Use a ruler. The line should extend slightly beyond the plotted points. Don't draw a tiny segment between just two dots.

Example 1 — Table of Values

Draw

y = 2 x + 1

for

- 1 \leq x \leq 2

Solution

Substitute each $x$ value to find $y$ .

$x$	$- 1$	$0$	$1$	$2$
$y$	$- 1$	$1$	$3$	$5$

Plot the four points and rule a straight line through them.

Example 2 — Intercept Method

Find the intercepts of

3 x - 4 y = 12

, then sketch the line.

Solution

$x$ -intercept: set $y = 0$ .

$3 x$	$=$	$12$
$x$	$=$	$4 \to (4, 0)$

$y$ -intercept: set $x = 0$ .

$- 4 y$	$=$	$12$
$y$	$=$	$- 3 \to (0, - 3)$

x = 4, y = - 3

Example 3 — Rearrange to y = mx + c

Rearrange

3 x + 2 y - 12 = 0

into slope-intercept form. State the slope and

y

-intercept, then sketch.

Solution

Solve for $y$ .

$3 x + 2 y - 12$	$=$	$0$
$2 y$	$=$	$- 3 x + 12$
$y$	$=$	$- \frac{3}{2} x + 6$

Slope $m = - \frac{3}{2}$ ; $y$ -intercept $= 6$ .

y = - \frac{3}{2} x + 6

Example 4 — Equation from a Graph

A line passes through

(0, 2)

and

(3, 0)

. Find its equation.

Solution

Find the slope using rise over run.

$m$	$=$	$\frac{0 - 2}{3 - 0}$
$m$	$=$	$- \frac{2}{3}$

$y$ -intercept is $2$ , so:

y = - \frac{2}{3} x + 2

y = - \frac{2}{3} x + 2

Common pitfalls

Only one point plotted. One point doesn't fix a straight line — plot at least two (three is safer, as a third checks for arithmetic errors).

Wrong axis for $x = k$ . The line

x = k

is vertical, not horizontal. The variable that's fixed is the one on the constant side:

x = 3

means "all points where

x

3

", which is a vertical line.

Drawing a segment, not a line. Use a ruler and extend the line slightly beyond the plotted points. A graphed line should look infinite, not like a tiny segment between two dots.

Forgetting to rearrange. If the equation is in the form

a x + b y + c = 0

, it's not yet in graphable form. Solve for

y

first (or use the intercept method directly).

Frequently asked questions

How do I draw the graph of a linear equation?

There are three common methods. Pick two or three x values, compute the matching y values, plot the points, and join with a straight line (table of values). Or find where the line crosses each axis and rule a line through the two intercepts. Or read off the y-intercept and use the slope to step to a second point.

What is the x-intercept and y-intercept?

The x-intercept is where the line crosses the x-axis. Find it by setting y equals 0 and solving for x. The y-intercept is where the line crosses the y-axis. Find it by setting x equals 0 and solving for y.

How do I graph y equals 5 or x equals 3?

y equals 5 is a horizontal line through every point with y coordinate 5, including (0, 5). x equals 3 is a vertical line through every point with x coordinate 3, including (3, 0). For y equals k the slope is 0, and for x equals k the slope is undefined.

How many points do I need to draw a straight line?

Two points are enough to define a line, but always plot at least three as a safety check. If all three lie on the same straight line you know your arithmetic is correct; if the third point doesn't line up, you have an error to fix.

How do I get an equation like 3x plus 2y minus 12 equals 0 into y equals mx plus c form?

Solve for y. Move all the x terms and constants to the other side, then divide every term by the coefficient of y. For 3x plus 2y minus 12 equals 0: 2y equals minus 3x plus 12, then y equals minus 3 over 2 times x plus 6.

How do I find the equation of a line from a graph?

Read off the y-intercept where the line crosses the y-axis (this is c). Use any two clearly identifiable points on the line to compute the slope m as rise over run. Then write the equation as y equals m x plus c.