Lesson Better Understanding Linear Equations and Slope

Source code of 'Better Understanding Linear Equations and Slope'

This Lesson (Better Understanding Linear Equations and Slope) was created by by Nate(3500)

Slope in Algebra is defined as M. The slope determines the 'tilt' of a line.
Equation: {{{M = (y2 - y1)/(x2 - x1)}}} where you have two points: (x1,y1) and (x2,y2)
Example:
What is the slope of the line through (0,1) and (1,2)
{{{M = (2 - 1)/(1 - 0)}}} x1 = 0 and y1 = 1 but x2 = 1 and y2 = 2
{{{M = (1)/(1) = 1}}}
You Can Reverse The Pair: (1,2) and (0,1)
{{{M = (1 - 2)/(0 - 1)}}} x1 = 1 and y1 = 2 but x2 = 0 and y2 = 1
{{{M = (-1)/(-1) = 1}}}
The Slope(M) = 1/1 or -1/-1
The numerator (top part of fraction) tells you how many units to go either up or down. If the number is positive, you go those units up. If the number is negative, you go those units down. The denominator (bottom part of fraction) tells you how many units to go either right or left. If positive, you go those units right. If negative, you go those units left.
Slope(M) = 1/1 one unit up and one unit to the right
Slope(M) = -1/-1 one unit down and one unit to the left
Find The Slope Of This Line:
Vertical Line: {{{x = -6}}}
Pick Any Two Points: (-6,0) and (-6,6)
{{{M = (y2 - y1)/(x2 - x1)}}}
{{{M = (6 - 0)/(-6 + 6)}}} x1 = -6 and y1 = 0 but x2 = -6 and y2 = 6
{{{M = 6/0}}} The Line's Slope Is Undefined
Find The Slope Of This Line:
Horizontal Line: {{{y = 3}}}
Pick Any Two Points: (0,3) and (-6,3)
{{{M = (y2 - y1)/(x2 - x1)}}}
{{{M = (3 - 3)/(-6 - 0)}}} x1 = 0 and y1 = 3 but x2 = -6 and y2 = 3
{{{M = 0/-6 = 0}}}
Here is the line:
{{{graph( 600, 600, -10, 10, -10, 10, x + 1 ) }}}
Let us look at point-slope form: {{{y - y1 = M(x - x1)}}} where a point: (x1,y1)
Find The Equation For The Linear Line Through (0,1) and (1,2)
We know: M = 1
We know: x1 = 0 and y1 = 1
{{{y - 1 = 1(x - 0)}}}
{{{y - 1 = x}}}
{{{y = x + 1}}} The Line
Let us look at slope-intercept form: {{{y = Mx + b)}}} where 'b' is the y-intercept
Find The Equation For The Linear Line Through (0,1) and (1,2)
When an x-term is equal to zero, the y-term tells you the y-intercept.
y-intercept = 1
Slope = 1
{{{y = Mx + b}}}
{{{y = x + 1}}}
Now, let us deal with inequalities:
greater than ( > )
less than ( < )
Graph: {{{y > 3x + 2}}}
From slope-intercept form we know:
M = 3 = 3/1 or -3/-1
Y-Intercept = 2
Since 'y' is only greater than the x-values, it is not also equal to so the line is dotted. Since 'y' is greater than the x-values, you would shade above the dotted line.
{{{graph( 600, 600, -10, 10, -10, 10, 3x + 2 ) }}}