Question

Note: This is NOT a question; it's a tutorial on linear equation. (I'm re-posting this because it disappeared from OpenStudy somehow..)

Answer 1

\[\large 1)~The~ General~form\]
\[Ax + By + C = 0\]
(Note here: I used capital A,B and C, because they are constants and they DO NOT represent ANY relevant information about the function. A & B must not be 0. Personally I hate this form.)
So yeah… You have nothing! No slope, no x-intercept, no y-intercept.

But formulas exist to find all of those.
To find the slope, use \[\frac{-A}{B}\]To find the x-intercept, use \[\frac{-C}{A}\]To find the y-intercept, use \[\frac{-C}{B}\]Let me give you an example. Let’s say we have the linear function [5x + 9y + 14 = 0\]
The slope would be \[\frac{-A}{B}=\frac{-5}{9}\]
The x-intercept would be \[\frac{-C}{A}=\frac{-14}{5}\]
The y-intecept would be \[\frac{-C}{B}=\frac{-14}{9}\]


\[\large 2)~The~ Slope-Intercept~form\]\[\large y=mx+b\]\[m=slope;~~b=y-intercept\]
What’s good with this form?
You have the y-intercept and the slope, what you need is the x-intercept.
To find the x-intercept, use \[\large \frac{-b}{m}\]
An example..: \[\large y = 45x + 8\]
The x-intercept would be \[\large \frac{-b}{m}=\frac{-8}{45}\]

BEWARE OF TRICKY QUESTIONS!
If we have something like \[\large y = 7(14x + 7)\]This is NOT the slope-intercept form, remember to DISTRIBUTE the 7 or any other constant to get it \[\large y = 7(14x + 7)\]\[\large y = 98x + 49\]


\[\large 3)~The~ intercept~form\](I love this one)\[\large \frac{x}{a}+\frac{y}{b}=1\]\[a = x-intercept~;~~~~b = y-intercept;~~~~~\frac{-b}{a}=slope.\]
See? You have everything here.


\[\large 4)~The~ point-slope~form\]
This is basically the formula to find a slope by using two points but slightly modified.
\[\large y - y_{1} = m(x-x_{1})\]\[m= slope; ~~~~x_{1} ~and~y_{1}: A~point~on~the~function\]
This form is *only* used to find the equation of a linear equation when the slope and a point is given.
Example, let's say we have a point at (-5,6) and a slope of 3, -5 and 6 are respectively x1 and y1.
\[\large y - y_{1} = m(x-x_{1})\]\[\large y - 6 = 3(x--5)\]\[\large y - 6 = 3(x+5)\]Distribute...\[\large y - 6 = 3x+15\]\[\large y = 3x+15+6\]\[\large y = 3x+21\]
We have the slope-intercept form!

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now.. How to find a equation of a linear function?

Situation 1: The slope and the y-intercept are given
Situation 2: The slope and the x-intercept are given
Situation 3. The slope and a random point of the function are given
Situation 4. Two random points of the function are given
Situation 5. The x-intercept and the y-intercept are given

>>>>>>>>>> Situation 1: The slope and the y-intercept are given

I give you... a slope! 8! and the y-intercept! 78!

Oh come on..
\[\large y=mx+b\]\[m=slope;~~b=y-intercept\]...\[\large y=8x+78!!!\]

Alright alright..

>>>>>>>>>> Situation 2: The slope and the x-intercept are given

I give you 19 as slope and 3 as the x-intercept
We know that \[\large 2)~The~ Slope-Intercept~form\]\[\large y=mx+b\]\[m=slope;~~b=y-intercept\]
To find the x-intercept, use \[\large \frac{-b}{m}\]Plug stuffs in...
\[\large \frac{-b}{m}=3\]\[\large \frac{-b}{19}=3\]\[\large \frac{-b}{19}*19=3*19\]\[\large -b=57\]\[\large b=-57\]
Back to the situation!

Slope 19, y-intercept -57
\[\large y = mx+b\]\[\large y = 19x-57\]

>>>>>>>>>> Situation 3. The slope and a random point of the function are given
I roll a dice.. I got 2 and 92, so our random point is at (2,92)
I roll a dice again... got 3..

D:

Point-slope form!
\[\large y - y_{1} = m(x-x_{1})\]\[m= slope; ~~~~x_{1} ~and~y_{1}: A~point~on~the~function\]
2 is x1, 92 is y1
\[\large y - y_{1} = m(x-x_{1})\]\[\large y - 92 = 3(x-2)\]\[\large y - 92 = 3(x-2)\]Distribute...\[\large y - 92 = 3x-6\]\[\large y = 3x-6+92\]\[\large y = 3x+86\]

>>>>>>>>>> Situation 4. Two random points of the function are given
Two random points. (19,31) and (22,16)
We know that m(the slope) is Rise/Run, or Delta y/Delta x, well, we are going use it to find the slope.\[m=\frac{16-31}{22-19}=-5\]
Okay, our slope is -5. We have a point, and a slope. Point slope form!
\[\large y - y_{1} = m(x-x_{1})\]\[m= slope; ~~~~x_{1} ~and~y_{1}: A~point~on~the~function\]
19 is x1, 31 is y1
\[\large y - y_{1} = m(x-x_{1})\]\[\large y - 31 = -5(x-19)\]Distribute...\[\large y - 31 = -5x+95\]\[\large y = -5x+95+31\]\[\large y = -5x+126\]

>>>>>>>>>> Situation 5. The x-intercept and the y-intercept are given
I give you x-intercept 1
and y-intercept 9

We are done. :)
\[\frac{x}{1}+\frac{y}{9}=1\]

Credits:
@lgbasallote
@AccessDenied
@amistre64
@shadowfiend

\(\underline{\qquad \qquad \qquad \qquad \qquad} \)

Answer 2

dont tell me you rewrote ALL that

Answer 3

No xD

Answer 4

I was smart enough to keep a \(\LaTeX\) version of the tutorial ;)

Answer 5

everyone is doing tutorials nowadays, someone should make a good one on integration and one on differential calculus where the graph of the derivative is considered. or a tutorial on advanced and complicated questions involving circular functions and logarithms

Answer 6

I shall try that out :D

Answer 7

Thanks zepp, it will really help me with my maths.

Answer 8

\[\Huge{\color{gold}{\star \star}\color{orange}{Great \space Job \space Dude \space ;)}}\]

Answer 9

⋆⋆Great Job Dude ;)