Equations of Straight Lines
When working with straight lines, there are several ways to arrive at
an equation which represents the line.
Slope is found by using the formula:
Slope is also expressed as
Remember:
rise/run.
Equation Forms of Straight Lines
Slope Intercept Form
Point Slope Form
Use this form when you know the slope and the yintercept (where the line crosses the y-axis).
Use this form when you know a point on the
line and the slope (or can determine the
slope).
y = mx + p
m = slope
p = y-intercept
(where line crosses the y-axis.)
m = slope
= any point on the line
Horizontal Lines
Vertical Lines
y = 3 (or any number)
x = -2 (or any number)
Lines that are horizontal have a slope of zero. Lines that are vertical have no slope (it does
Horizontal lines have "run", but no "rise". The not exist). Vertical lines have "rise", but no
rise/run formula for slope always yields zero
"run". The rise/run formula for slope always
since the rise = 0.
has a zero denominator and is undefined.
Since the slope is zero, we have
y = mx + p
The equations for these lines describe what is
y = 0•x + 3
happening to the x-coordinates. In this
y=3
example, the x-coordinates are always equal to
This equation also describes what is happening
to the y-coordinates on the line. In this case the
-2.
y-coordinates are always 3.
Examples:
Examples using Slope-Intercept Form:
1. Find the slope and y-intercept for the
equation 2y = -6x + 8.
1st solve for "y" by dividing both sides by
2 -> y = -3x + 4
Remember the form: y = mx + p
Answer: the slope (m) is -3
the y-intercept (p) is 4
Examples using Point-Slope Form:
3. Given that the slope of a line is -3 and the
line passes through the point (-2,4), write the
equation of the line.
The slope: m = -3
The point (x1 ,y1) = (-2,4)
Remember the form: y - y1 = m ( x - x1)
Substitute:
y - 4 = -3 (x - (-2))
y - 4 = -3 ( x + 2)
If asked to express the answer in "y =" form:
y - 4 = -3x - 6
y = -3x - 2
2. Find the equation of the line whose slope 4. Find the slope of the line that passes
is 4 and the coordinates of the y-intercept are through the points (-3,5) and (-5,-8).
(0,2).
In this problem m = 4 and p = 2.
First, find the slope:
Remember the form: y = mx + b=p
and
that p is where the line crosses the y-axis.
Substitute:
y = 4x + 2
Use either point: (-3,5)
Remember the form: y - y1 = m ( x - x1)
Substitute: y - 5 = 6.5 ( x - (-3))
y - 5 = 6.5 (x + 3)
Gradients of parallel and perpendicular lines
On a graph, parallel lines have the same gradient.(slope)
* For example,
and
are parallel because they both have a gradient of 2.
* From a graph,
All the points that lie on the green line have a coordinate that is the same as the
coordinate
For example, the points (-1,-1) and (2,2) belong to this line.
We say that the equation of the line is
All the points that lie on the pink line have a coordinate (the second number in brackets) that is one
number higher than the coordinate of the same line. For example, the points (-3,-2) and (0,1) belong
to this line.
In other words, the coordinate equals the coordinate
.So the equation of the line is
Both lines have a slope (gradient) equal to 1. These are parallel lines
.
Remember that perpendicular lines will always cross at right angles.
In this diagram, the lines
and
cross at right angles.
The gradients of these lines are 2 and
.
You can work out whether 2 lines are perpendicular by multiplying their gradients. The product of the
gradient of perpendicular lines will always be -1. If lines are perpendicular, m1 m2 = − 1
To find the equation of a line that is perpendicular to a given line, you will need to work out the
gradient of one line before finding the gradient and equation of the other.
Find the perpendicular line to
Re-arrange the equation
through the point (0, 2).
in the form
The gradient is …
Now we need to work out the gradient of the 2nd line. Remember that when 2 lines are perpendicular
the product of their gradients is
. Let's call the gradient of the second line m.
so m’ = .....
In the question we are told that the line passes through the point (0, 2). This means that the line crosses
the y axis at
.
So the equation of the line that is perpendicular to
is y = ….