INTRO: When you have a situation where you are comparing two quantities and the one quantity is increasing or decreasing in a constant manner with respect to the other quantity, you can describe it with a linear equation, and graph it with a line.

In general it is easier to write these equations if you think of the two quantities as variables and give them letters that match what they stand for, like let "a" = the number of ants, and "g" = the number of grasshoppers. Then decide which variable is the dependent variable and which is the independent variable. You have to read the problem or look at the data carefully to decide which quantity depends upon the other.

The independent variable is always graphed on the "x" axis and the dependent variable on the "y" axis. Remember, an equation written in "slope-intercept" form,
y=sx+i, is easy to graph. The "s" is the slope or the amount of change and "i" is what you start with before you even change at all. Sometimes "i"=0.

If you can find that equation, you can answer lots of "what if" slope scenarios.

Let's take a look at an example:

Tulip shown here picking her apples.

Chris and his sister Tulip are picking apples for the annual apple dumpling making extravaganza held in their kitchen each fall. Chris can pick three apples for every one that Tulip picks. Find an equation to compare the amount that he picks to the amount that she picks. ***********

The amount Chris picks is three times more than the amount Tulip picks. So if
"c" = the amount that Chris picks and "t" = the amount that Tulip picks, one equation would be c = 3t. Here the independent variable is the "t" and the dependent variable is the "c". You see, no matter how many apples Tulip picks, "t", Chris will always pick three times more.

The slope here is three because that is the amount of increase for every apple that Tulip picks. Notice the y-intercept here is zero.

Now play "WHAT IF":

If Chris has 102 apples in his pile at the end of the day, how many will Tulip have?

Solution: Substitute 102 for Chris' amount, "c", into the equation c=3t, then solve for Tulip's amount, "t" .

102 = 3t

102/3 = 3/3t ( divide both sides of the equation by 3)

34 = t

So Tulip has 34 apples in her pile. You can check your answer by taking her amount, 34, and multiplying it by three. 34 times 3=102 so it checks.

If Tulip has 78 apples in her pile how many does Chris have?

Solution:

c = 3(78)

c = 234 apples

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p=5+d

r

d=5p

t

p=5d

l

11.) Jenna is writing a book of "Quizzles". She plans on becoming rich from her unique book of algebraic thought problems. On page 32 we read, "The sum of three, consecutive, even, integers is equal to 'y'."

Write a linear equation in "slope-intercept" form, that lets the sum be the dependent variable and the smallest integer be the independent variable.

13.) Zack is working for the Grinch on a cleanup project. For every ton of garbage he removes, the Grinch will pay him $13.95.

Write an equation in y=mx+b form where the independent variable is tons of garbage and the dependent variable is dollars earned.

16.) Erik is in training to become a standup comedian. He is taking lessons at a local school named, "Choose Your Comedy Like Your Life Depended on It Academy for the Humorously Gifted".

He had to pay some initial costs: a $75 registration fee, and he had to buy a copy of "The Book of Comedy", at $65.98. His classes cost $7.50 per hour.

Write an equation in "slope intercept" form where Erik's total cost of education depends upon the hours spent in class and his initial costs.