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![Page 1: Presented By: Katie Graves. In many daily instances, two quantities are related linearly. This means that a graph of their relationship takes the.](https://reader036.fdocuments.in/reader036/viewer/2022062906/5a4d1b517f8b9ab0599a7bf0/html5/thumbnails/1.jpg)
Presented By: Katie Graves
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In many daily instances, two quantities are related linearly. This means that a graph of their
relationship takes the form of a straight line.
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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
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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.
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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
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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.
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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
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How would we graph: 4y + 2x = 12 ? What are our 2 points?
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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).
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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
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For example, if we were to graph y=mx+b, it would look like:
![Page 12: Presented By: Katie Graves. In many daily instances, two quantities are related linearly. This means that a graph of their relationship takes the.](https://reader036.fdocuments.in/reader036/viewer/2022062906/5a4d1b517f8b9ab0599a7bf0/html5/thumbnails/12.jpg)
Y=(-4/3)x+3 How do we graph this?
Y=2x+6 How do we graph this?
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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)
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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
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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
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What would the the slope be for the points… (5,6) and (8,9)
(1,4) and (3,7)
(4,4) and (0,4)
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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!
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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!
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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
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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
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Using the Calculator Rangers, we are going to attempt to “Match the Graph”
Would anyone like to volunteer to try it out first?