standard form for an algebraic equation?

Question

When dealing with linear relationships, what are the advantages of using a standard form for an equation. We are studying point-slope, standard form, and explicit and recursive. We are expected to be able to convert each of them. I'm just unsure of the advantages for using each one.

Answer 1

Standard form is great to find the x and y intercepts.
For example in 3x+4y=12 (standard form)
I quickly find the x intercept by plugging in y=0
3x+4(0)=12
so 3x=12
x=4 when y=0
4,0 is my x intercept

Similarly the yintercept is found by plugging in x=0
3(0)+4y=12
4y=12
y=3
(0,3) is the y intercept

Slope intercept is easier for graphing
Given an equation like y=2/3x+2
You make a point at (0,2)
Then you count rise 2 run 3 to make your other points.
Connect the dots and you can quickly graph the line.

I hope this helps!

As far as Explicit versus recursive, explicit is used to find a value given any x. The recursive is about finding the next value given the value before.
In f(x)=3x+2 the f(0)=2, f(1)=5,f(2)=8 so f(n)=f(n-1)+3
This is good for showing how a series of values exists based on a common difference with each answer based on the answer before.

Answer 2

y = mx+b is my favorite because it immediately tells you the slope and y-intercept. It is also handy if you are trying to find the intersection of two lines, since you just set the mx+b of one = to mx +b of other and solve for x.

The general or standard form is ax+by+c =0. This could be handy if you are using matrices to solve.

Point-slope form is y-y1 = m(x-x1), which is efficient if you are given the slope and a point (x1,y1) other than the y-intercept.

The intercept form x/a +y/b = 1 gives you the equation immediately if you know the x- and y- intercepts (a and b respectively).

The two-point form is (y-y1)/(x-x1) = (y2-y1)/(x2-x1) gives you the equation if you are given two points (x1,y1) and (x2,y2).

In the long run, pick whichever you are most comfortable with. It is valuable to know the different forms available to you.