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Equations of a Line
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Gradient-intercept form, point-gradient form, vertical lines, horizontal lines, standard form of a line
Want a deeper conceptual understanding? Try our interactive lesson!
Gradient-intercept form
SL Core 2.1
A straight line is defined by its gradient and its y-intercept. The gradient-intercept equation of a line is thus:
y=mx+c📖
Point-gradient form
SL Core 2.1
If we know a point (x1,y1) on a line and the gradient m of the line, we can use the point-gradient form of the line:
y−y1=m(x−x1)📖
Try playing around with this diagram and seeing how changing the values of x1,y1, and m impacts the graph:
Vertical lines
SL Core 2.1
A vertical line does not have a well defined gradient, since there is no "run" - the x-values never change.
We cannot write the equation of a vertical line in the form y=⋯. Instead we write
x=k
for some constant k.
Horizontal lines
SL Core 2.1
A horizontal line has gradient m=0. It is therefore in the form
y=c
for some constant c.
Standard form of a line
SL Core 2.1
The equation of a straight line can also be given in the form
ax+by+d=0📖
This reduces to
y=−bax−bd
In examinations, you may be asked to write the equation of a line in standard form.
Try playing around with this diagram and seeing what you find:
Here are some example equations converted from one form to another:
It's important to note that standard form has the tightest restrictions on how an equation can be written. With the other forms, we can write the same equation in a couple of different ways, depending on the point we choose or fractions we want to work with.
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