Linear Equations Reference Sheet
www.apexmath.com.au
Equation of a Line
Gradient (Slope)
Important Formulas
Gradient
-
Intercept
Form
– gradient (slope)
– y-intercept
Useful for interpreting the
relationship between variables x
and y, graphing, and identifying
m and b. Also useful for
generating the equation of the
line when m and b are given (ie
from a graph).
Point
-
Gradient
Form
, – a point on the line
– gradient (slope)
Useful for plugging in a point on
the line (once the gradient is
known) to generate the
equation of the line. It may then
be changed into another form.
Standard Form
is non-negative and ,,
are relatively prime integers
(no common factors)
Useful for finding both the x-
and y-intercepts of the line and
using the intercepts to graph
the line.
Horizontal Line
All points have y-coordinate
The line is horizontal with gradient
0
and y-intercept
.
Vertical Line
All points have x-coordinate
The line is vertical with an undefined
gradient.
Gradient
Formula
rise
run
, & , – points on the line
The formula is “rise over run”, which is
the change in y divided by the change in
x.
Reading Gradient
From a Graph
It is recommended to read the gradient from
left to right. If the line goes up from left to right,
the gradient is positive. If the line goes down
from left to right, the gradient is negative.
Parallel Lines
The gradients of parallel lines are equal.
Parallel lines never intersect.
Perpendicular Lines
1
The gradients of perpendicular lines are
negative reciprocals of each other.
Perpendicular lines intersect at 90°
(right) angles.
Mid
-
Point Formula
,
2,
2
, & , – coordinates
The formula can be broken up into two
parts, the x-part and y-part. Each part is
the middle (or average) of the x and y
coordinates of the given points.
Distance Formula
, & , – coordinates
The formula is derived from the Pythagorean
Theorem . The legs of a right
triangle are given by the difference in x- and
y-coordinates (inside the brackets). The
distance is the hypotenuse of this right
triangle. This is calculated by the square root
of the sum of the squares of the legs (right-
hand-side of formula).
Perpendicular Distance
|
|
√
Given an equation in the form
0 and a point ,
The equation of the line must be in the above
form and the perpendicular (shortest)
distance from the point to the line can be
calculated by plugging in all values.
General
Form
0
is non-negative and ,,
are relatively prime integers
(no common factors)
Similar to standard form
except it is solved for 0 (all
terms on left side). Not very
useful for knowing anything
about the line without
formulas. Recommend
converting to another form.