Algebra 2 Variation Word Problems
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Direct Variation
Variation Word Problems

 

 It would be great if every time we had to solve a problem,

 somebody just gave us the equation to use.
 Well guess what,
 that doesn't always happen.
 Lots of the time we get a little story that 
 we have to translate into an equation.
 
 This whole thing usually starts with two values that change.
 Usually one of the things can change on its own
 and the other thing changes BECAUSE the first thing changed.
 That is, if the first thing hadn't changed, 
 the second thing wouldn't have changed.
 
 The thing that can change by itself 
 is called the independent variable.
 It does what it wants, it's INDEPENDENT.
 The thing that changes BECAUSE the first thing changed 
 is called the dependent variable.
 What it does depends on what the first thing does,
 it is DEPENDENT on the first thing.
 
 In math, the thing that can do what it wants is often called X.
 and the thing that changes depending on what X does is often called Y.
 
 Sometimes when X gets bigger so does Y.
 
 If when X gets bigger Y gets bigger too,
 it is called direct variation.
 In math talk we say "Y varies directly as X."
 In English we say "If X goes up, so does Y. If X goes down, so does Y."
 
 If Y varies directly as X and they both have the same value
 the equation for this is:

 Y = X

 
 Even if X and Y are both going up or down, 
 They usually do not have the same value .
 
 Example:
 If I am traveling down the road at 50 miles per hour,
 after 1 hour, time has changed by 1 hour.
 but miles traveled has changed by 50 miles.
 
 We can change our Y = X (varies directly) equation
 to show this relation.
 As X changes by 1 Y changes by 50,
 so we just need to multiply X.
 
 We can fix our Y = X equation to do this.
 We make it:

 Y = 50X

 
 In general, we call this form:

 Y = kX

 Where k stands for any number.
 
 Example:
 Y varies directly as X.
 When Y = 2, X = 10.
 Find k ...

 

 
 So for this problem, the equation is ...

 

 
 Example:
 For the problem in example 1, find Y when X = 25.

 

 
 One thing you can probably see is that in this problem, 
 when X = 0 then Y = 0 too.
 Sometimes "Y varies directly as X", But when X = 0, Y = not zero.
 This is almost the same thing, but leave it to math people
 to give it it's own special name.
 They say:
 "Y varies linearly with X"
 In general, the equation looks like:
 

 Y = kX + C

 
 We have 4 things here, X, Y, k, and C.
 To solve this one, we need to know a lot ...
 
 Example:
 Y varies linearly with X. When X = 0, Y = 5. When X = 3, Y = 11.
 Find the entire equation.
 
 OK, taking one piece at a time ...

  Y = kX + C

 
 When X = 0, Y = 5 so ...

  Y = kX + C

 5 = k0 + C

 5 = C

 
 So now we have ...

  Y = kX + 5

 
 We still need to find k.
 We are told that when X = 3, Y = 11.
 So put those numbers in and see what you get ...
 

 

 
 So we have:

  Y = 2X + 5

 
 Example:
 Y varies linearly with X.
 When X = 2, Y = 15.
 When X = 4, Y = 23
 
 OK, here we don't know the value of Y when X = 0
 so we need more tricks ...
 What we do is put in the first X and Y values and get one equation,
 Then put in the second X and Y values and get a second equation.
 
Y = kX + C Y = kX + C
Y = 15, X = 2 Y = 23, X = 4
15 = k(2) + C 23 = k(4) + C
15 = 2k + C 23 = 4k + C
 
 What we have now is 2 equations with 2 unknowns...
 The only weird thing is that the unknowns are k and C
 rather than X and Y.
 But k and C and X and Y are just names.
 It really doesn't make any difference to the math
 if you use an X or a Q or a J or whatever ...
 
 OK, so 2 equations with the same two unknowns.
 We can rewrite both equations as C = stuff
 and then substitute the stuff in 1 equation
 for the C in the other.
 Then we will have an equation with 1 unknown.
 That we can solve!
 Here goes ...

 

 
 Now find C ...

  15 = 2k + C

  15 = 2(4) + C

  15 = 8 + C 

 7 = C

 
 So C = 7 and k = 4.
 That means the equation is ...
 

  Y = kX + C

 Y = 4X + C

 

   copyright 2005 Bruce Kirkpatrick

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