Presented By: Katie Graves

In many daily instances, two quantities are related linearly. This means that a graph of their

relationship takes the form of a straight line.

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

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.

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

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=8Therefore, when y = 0, x=8. The point with coordinates (8, 0) lies

on the line.

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

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

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).

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

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

Y=(-4/3)x+3 How do we graph this?

Y=2x+6 How do we graph this?

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)

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 soldTemperature: C = (5/9) (F-32) This equation shows how to convert

Fahrenheit Temperatures (F) to Celsius (C) Temperatures

How do we find slope? When given two points (x1, y1) and (x2,

y2) we use the form: y2-y1

x2-x1 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

What would the the slope be for the points… (5,6) and (8,9)

(1,4) and (3,7)

(4,4) and (0,4)

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!

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!

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’)

-5 0 5 10

15

20

Wind Chill (F’) -45

-38

-31

-10

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

Using the Calculator Rangers, we are going to attempt to “Match the Graph”

Would anyone like to volunteer to try it out first?