Taming the algebraic duplicity of lines!
Lines are represented by linear equations. But there are a couple of options in how to present such linear equations, giving us two quite different ways of thinking about how the equation of a line relates to its geometrical form in the Cartesian plane.
In this lesson we will learn

about the two different algebraic forms for a line

the significance of the slope and the intercepts of a line.
Two ways of writing a straight line
Every line can be written in the form
for some numbers . In fact we should require that and are not both zero. This form has the advantage of being the least restrictive.
Every nonvertical line can be written in the form
This is called the slopeintercept form of a line. This form emphasises the slope , also called the gradient , along with the intercept of the line.
Q1 (M): What line cannot be written in slopeintercept form?
So we have seen two different ways of writing the same line. For example the line can also be written as . The line can also be written as .
Q2 (E): What is the slope of the line ?
Q3 (E): Find the  and intercepts of the line pictured below.
Manipulating general equations
It is often instructive to work with abstract equations rather than special cases. Having letters represent variable quantities, such as the coefficients of and in the equation of a line can lead to more insight into what is really going on.
For example if we multiply two numbers, say and , then the result is , but the original two numbers are more or less invisible. However if we multiply two variables, say and , then the result is and we can see more clearly where this came from.
So the general line
can be rearranged first as and, if , then as
Going the other way, could be rearranged as .
Q4 (M): Find the slope and  and  intercepts of .
Q5 (M): Find the  and  intercepts of .
Answers
A1. The line with equation cannot be written in the form.
A2. The line can also be written as
We can read off the slope from this form of the line: .
A3. To find the intercept of the line we set . This gives the equation so that . To find the intercept we set , which gives the equation so that . Have a look at the picture of the line to convince yourself that these values are correct.
A4. The slope of this line is . The intercept of is what you get when you set and solve for , namely . The intercept is what you get when you set and solve for , namely .
A5. The intercept of is and the intercept is . Note that the intercept is the easy one to find.
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