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Solving Equations Solving Equations and Formulasand Formulas
Chapter 3 section 8Chapter 3 section 8
Solving for a specific Solving for a specific VariableVariable
Sometimes Sometimes equations involve equations involve multiple multiple variables, or variables, or letters that stand letters that stand for specific for specific things, such as things, such as scientific scientific formulas and formulas and physics equationsphysics equations
Geometry Formulas:Geometry Formulas:
The formula for The formula for Circumference of a Circumference of a circle is:circle is:
C=2πr (2 x pi x radius)
We can solve the formula for r: (this means get r by itself!)
What do you think we could What do you think we could do?do?
If the equation is set up as If the equation is set up as C = 2πr, and you are asked to get r by itself, then you need to first ask yourself: What is attached to r? How can you “undo” the operation?
Since 2π is “stuck” to r by multiplication, you could get rid of it by division……
Like this!Like this!
C =2πr Divide both sides by 2π: 2π 2π This leaves you with the following:
C = r2π
That’s it! That is all there is to it! Now r is by itself, and all of the other “stuff” is on
the other side!
Here’s another one:Here’s another one:
Try solving the following formula for x:Try solving the following formula for x:
5x + y = x5x + y = x
Follow these steps to get Follow these steps to get x:x:
5x + y = x5x + y = x1.1. Move your x’s together, to the same side: Move your x’s together, to the same side:
Which one should you move? The 5x or the x???Which one should you move? The 5x or the x???
5x + y = x5x + y = x - 5x -5x- 5x -5x
(you should move the 5x because it needs to be with (you should move the 5x because it needs to be with the x on the other side, and it needs to be away the x on the other side, and it needs to be away from the y!from the y!
Steps 2 & 3:Steps 2 & 3:
Now your equation should look like this!Now your equation should look like this!
y = -4xy = -4x
2. What would be the next thing you should do to get the x by 2. What would be the next thing you should do to get the x by itself? itself?
(If you said DIVIDE BY – 4, THEN YOU’RE RIGHT!)(If you said DIVIDE BY – 4, THEN YOU’RE RIGHT!)
yy = = -4x-4x-4 -4-4 -4
3. Your answer will be 3. Your answer will be yy = x = x -4-4
How ‘bout How ‘bout another one?!?another one?!?
Solve 6 – ay = 4(a – b) for a
Just follow the steps you have Just follow the steps you have been using in other types of been using in other types of
equations!equations! 6 – ay = 4(a – b) Distributive property:
6 – ay = 4a – 4b Get the things with an “a” together, and the
things that don’t have an “a” together:
6 – ay = 4a – 4b - 4a - 4a Simplify to get:
6 – ay – 4a = - 4b
Big Finish:Big Finish:
6 – ay – 4a = - 4b -6 -6 Simplify to get: - ay – 4a = - 4b – 6 Now, here’s the tricky part: You have two things on
the left that have a’s and you can’t combine them because they aren’t like terms!
You have to do something called Factoring: it goes like this: Ask yourself what a is being multiplied by in the two terms on the left: there is an a with – y, and an a with – 4, so you group them in parentheses:
Factor:Factor:
- ay – 4a = - 4b – 6Now looks like this: a(- y – 4) = - 4b – 6
(Its like the distributive property, backwards!) To get the a by itself, divide both sides by the
stuff in the parentheses: a(- y – 4) = - 4b – 6 (- y – 4) (- y – 4)
Your answer looks REALLY WEIRD!, but that’s OK!
Your answer should look like this:Your answer should look like this:
a= - 4b – 6 - y – 4
(you don’t have to keep the () on bottom any more)
How’s that for fun!?!
Try one more :Try one more :(if your brain is not already (if your brain is not already
fried!)fried!) The perimeter of a square field is
given by the equation P = 2l + 2w, where P represents the
perimeter, l represents the length of the field, and w represents the width of the field.
Solve the formula for l.
What should you do first?What should you do first?
Answer:Answer: P = 2l + 2w - 2w - 2w
P - 2w = 2l 2 2 P - 2w = l 2
OMG!!!!OMG!!!! Yes, these are hard!Yes, these are hard! Yes, you have to do them!Yes, you have to do them! Yes, you have an Yes, you have an
assignment!assignment! Book page 168 14 – 32 evenBook page 168 14 – 32 even
(10 problems)(10 problems)
You’ll thank me one You’ll thank me one day…….day…….