SOLUTION: What 2 things do we need to write the equation of a line ?

Question 613594: What 2 things do we need to write the equation of a line ?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you need a minimum of either:
(a) 1 point on the line and a slope
(b) 2 points on the line
from 2 points on the line, you can calculate the slope and then, using one of the points, can determine the equation.
equation for a straight line is:
(a) standard form:
ax + by = c
(b) slope intercept form:
y = mx + b
(c) point slope form:
y-y1 = m(x-x1)
m is the slope
b is the y-intercept
m can be determine by the equation of:
m = (y1-y2)/(x1-x2)
(x1,y1) is one point on the line.
(x2,y2) is the other point on the line.
once you know the slope, you can use either the point slope form or the y intercept form to generate the equation of the line.
example:
2 points are (1,5) and 2,7)
assign (1,5) to be (x1,y1) - you can assign either one of the points - it doesn't matter.
assign (2,7)) to be (x2,y2) - this is the other point.
from those 2 points, find the slope.
m = (y1-y2)/(x1-x2) = (5-7)/(1-2) = -2/-1 = 2
slope intercept form of the equation becomes:
y = 2x + b
substitute either point for x and y and solve for b.
b is the y-intercept.
example:
use the point (2,7)
slope intercept form of the equation becomes:
7 = 2(2) + b which becomes:
7 = 4 + b which becomes:
b = 3
slope intercept form of the equation becomes:
y = 2x + 3
you could also have used the point slope form of the equation once you knew that m = 2
point slope form of the equation is y-y1 = m(x-x1) which becomes:
y-y1 = m(x-x1)
assign one of the points to (x1,y1) - it doesn't matter which one.
we'll use (1,5) = (x1,y1)
point slope form of the equation becomes:
y-5 = 2(x-1)
we can now convert this to slope intercept form by solving for y as follows:
add 5 to both sides of the equation to get:
y = 2(x-1) + 5 which becomes:
y = 2x-2 + 5 which becomes:
y = 2x + 3
same equation as before only we derived it a different way./
to get to the standard form of the equation, start from the slope intercept form of the equation and manipulate to the standard form as shown below:
y = 2x + 3
subtract 2x from both sides of the equation to get:
-2x + y = 3
you can leave it like this or you can multiply both sides of the equation by -1 to make the x term positive.
if you do, then you get:
2x - y = -3
either form is acceptable.
all forms of the equation for the same straight line can be derived from either form with a little algebra.

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