1. Presented By: Katie Graves

2.  In many daily instances, two quantities are related linearly.  This means that a graph of their relationship takes the form of a straight line.

3.  If x and y are two variables, and a, b and c are constants, then an equation relating x and y which takes the form:  ax + by = c  The following are linear equations because they are of this form:  3x + 2y = 7  4x − 8y = 2  −2x + y = 9

4.  Any straight line graph can be drawn by plotting just two points which satisfy the linear equation and then joining them with a straight line.

5.  Our Goal: To find two points which satisfy the equation 3x + 4y = 24 and hence plot its graph.  To find Points…  First, set x = 0. This will give us the y-intercept.  Then, we have 3(0) +4Y=24  This is the same as 4Y=24.  When we divide both sides by 4, we have Y=6  When x = 0, y = 6.  Therefore, we know that (0, 6) lies on the line

6.  Next, we set y = 0.  Then our equation becomes 3x+4(0)=24  From there, we obtain 3x=24  Dividing 3 from both sides, we end up with x=8  Therefore, when y = 0, x=8.  The point with coordinates (8, 0) lies on the line.

7.  Next, we plot our points (0,6) and (8,0)  After plotting the two points, we join them together with a straight line  We must note that the line slopes downwards as we move from left to right

8.  How would we graph: 4y + 2x = 12 ?  What are our 2 points?

9.  When set y=0, we get 2x=12. Dividing both sides by 2, we obtain x=6. Our point is (6,0)  When we set x=o, we get 4y=12. Dividing both sides by 4, we obtain y=3. Our point is (0,3).

10.  While, linear equations can be written in the form: ax+by=c, they are more commonly written as: y=mx+b  From this equation, we can figure out slope very easily, as well as the y-intercept (where the line goes through the y-axis)  m=slope  b=y-intercept

11.  For example, if we were to graph y=mx+b, it would look like:

12.  Y=(-4/3)x+3  How do we graph this?  Y=2x+6  How do we graph this?

14.  y=mx+b can be used for many real life applications:  Calories Burned in a Workout: y=215+3.8x ▪ When you start the workout you’ve already burnt 215 calories ▪ Each minute, 3.8 additional calories are burnt (x=minutes)  Earnings from Mowing a Lawn: y=-300+15x ▪ Buying a mower cost $300 ▪ You earn $15/lawn mowed (x=number of lawns)

15.  Income as a Waitress: y=45+.15x  Each day working, you earn $45  You also earn 15% (or 15 cents) of each dollar of food sold  Temperature: C = (5/9) (F-32)  This equation shows how to convert Fahrenheit Temperatures (F) to Celsius (C) Temperatures

16.  How do we find slope?  When given two points (x1, y1) and (x2, y2) we use the form: y 2 -y 1 x 2 -x 1  For example, if we have the points (3,1) and (2,6), our slope would be: (6-1) (2-3)  This is equal to 5/-1 or -5

17.  What would the the slope be for the points…  (5,6) and (8,9)  (1,4) and (3,7)  (4,4) and (0,4)

18.  When equations have a positive slope, the “y” increases from left to right  What is the slope of this equation?  Reminder: Slope is rise over run!

19.  When a slope decreases, the “y” values decrease from left to right  What is the slope of this graph?  Reminder: Slope is rise over run!

20.  Meteorologists often give both the actual temperature and the wind chill  Here is an example for when the wind speed is 20 mph  Do you see a pattern? What do you think the missing temperatures are? How could we make a linear equation for this table? Temperature (F’) -505101520 Wind Chill (F’)-45- 38 - 31 - 10

21.  When the numerator on the “rise over run” equation equals zero, we have a slope of zero  For example, the equation for this graph is y=2

22.  Using the Calculator Rangers, we are going to attempt to “Match the Graph”  Would anyone like to volunteer to try it out first?