Algebra Period 3

COMPARING THE 3 FORMS OF LINEAR EQUATIONS

Standard Form
Ax + By = C
3x + 4y = 10 is the STANDARD FORM of a line
x and y are on the same side of the equation and both coefficients are integers.
This format works especially well when the coefficients are both factors of the constant.
Use the x and y intercepts to graph.

Slope Intercept Form
y = mx + b
y = -3/4 x + 5/2 is SLOPE INTERCEPT FORM of the same line
y is isolated on one side, x term is first, then the constant on the other side of the equation
The coefficient of the x term is the slope.
The constant is the y intercept.
Graph the y intercept, then count the slope to another point.
For graphing, it doesn't work well if the y intercept is a fraction!



Point Slope Form
y - y1 = m(x - x1)
y - 3 = 3/4(x - 4)
Works really well if you have the slope and a point on the line.
You can use this method, and then simplify to the point slope form.

FITTING EQUATIONS TO DATA: 7-7
(word problems)
Many real world relationships are LINEAR, meaning they can be graphed with a LINE.
For example, if candy bar costs $1.50, then 2 bars cost $3.00 etc
Think of the number of candy bars as the x value because that's what you decide
(how many candy bars you're going to buy)
The result will be how much money you owe at the register (the y value)
x y
1 --> $1.50
2 --> $3.00
3 --> $4.50
If you graph this, you'll get a line because the price is constant.
That means that the slope is constant.
What is the slope????
The price = 1.50
Think of it as the change in y ($ you owe) over the change in x (# of candy bars)
The money you owe goes up $1.50 every time you buy 1 more candy bar
This is a POSITIVE slope of 1.50
So another meaning of a positive slope is two types of data that GO IN THE SAME DIRECTION
You can reverse both directions as well:
You don't have enough money for 3 candy bars, so you decrease your purchase by 1 bar
Then your purchase price also decreases by $1.50
The data is still going in the SAME DIRECTION (both now going down!)

If we now what to find the equation of this linear relationship, simply use the slope intercept or point slope formulas of a line!

SLOPE INTERCEPT:
y = 1.5x
(the y intercept is 0 because at 0 candy bars, you owe 0)
POINT SLOPE:
y - 1.50 = 1.50(x - 1)

Y INTERCEPTS IN THE REAL WORLD THAT ARE NOT ZERO:
The scenario above has a y intercept value of 0 because you don't owe anything if you don't buy anything, But often, the y intercept value will be a number. For example, think of cell phone use. Say you are charged $.10 per minute of use, but your monthly charge is $25.
Even if you don't use your cell phone, you still owe $25!
The linear equation would be:
y = .10x + 25 where y = what you owe and x = number of minutes used
Plug in 0 for x, number of minutes, and you still owe $25 (y value)

Let's think of a real life example that will give us a NEGATIVE SLOPE...
Often, the more you buy, the smaller the unit price per item.
This happens with copying or buying things like invitations.
Companies give you a "break" if you buy more.
SEE THE EXAMPLE ON PAGE 333 IN YOUR BOOK!
(we'll go over this one in class)

Finally, sometimes real world data can be APPROXIMATED as a linear relationship.
In other words, it may not be exact, but a good way to understand the data is to look at it that way.
Think SCATTER PLOTS with POSITIVE or NEGATIVE CORRELATIONS.

2 SPECIAL LINES AND THEIR SLOPES: 7-8

PARALLEL LINES:
2 lines that are parallel to each have the SAME SLOPE!
y = 2x - 10 and y = 2x + 3/4 are parallel because they both have a slope of 2


PERPENDICULAR LINES:
2 lines that are perpendicular to each other have SLOPES that are:
OPPOSITE SIGNS and RECIPROCALS
y = 2x - 10 is perpendicular to y = -1/2 x + 3/4

THIS IS ANOTHER TWIST TO OUR MYSTERY LINE PUZZLE!!!
If you know that the mystery line is parallel or perpendicular to another given line
then you know the mystery line's slope!!!

EXAMPLE:
Your mystery line has a point of (2, -5) and is PARALLEL to line y = 2x + 3/4
So you know the mystery line's slope because it is the same as the given line ( m = 2)
Substitute the slope and the point given on the mystery line and solve for b.

EXAMPLE:
Your mystery line has a point of (2, -5) and is PERPENDICULAR to line y = 2x + 3/4
So you know the mystery line's slope because it is the opposite sign reciprocal of the given line
(since the given line's slope is 2, the mystery line's slope is -1/2)
Substitute the slope and the point given on the mystery line and solve for b.