solving equations 2013-02-21 1 slide per page

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PHYSICSSolving Equations

www.njctl.org

April 2012
http://www.njctl.org/http://www.njctl.org/http://www.njctl.org/


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Our goal is to be able to solve any equation for any variablethat appears in it.

Let's look at a simple equation first.

The variables in this equation are s, d and t.

Solving for a variable means having it alone on the left side.

This equation is currently solved for "s".

Solving for a Variable

d

t

s =


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Like in any game there are a few rules.

1. To "undo" a mathematical operation, you must do the opposite.

2. You can do anything you want (except divide by zero) to oneside of an equation, as long as you do the same thing to theother.

3. If there is more than one operation going on, you must undothem in the opposite order in which you would do them, theopposite of the "order of operations."

4. You can always switch the left and right sides of an equation.

The Rules


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Let's solve this equation for "d"

That means that when we're donewe'll have d alone

on the left side of the equation.

The Rules

d

t

s =


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1 First, is d already ALONE? If not, what is with it?

A s

B d

C t

D it is already alone

dt

s =


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2 What mathematical operation connects d and t?

A d is added to t

B d is multiplied by t

C d is divided by t

D t is subtracted from d

dt

s =


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3 What is the opposite of dividing d by t?

A dividing t by d

B dividing by s into t

C multiplying d by t

D multiplying by t by d

dt

s =

Rule 1. To "undo" a mathematical operation, you must do theopposite.


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4 What must we also do if we multiply the right side by t?

A divide the left side by t

B multiply the left side by t

C divide the left side by dD divide the left side by d

dt

s =

Rule 2. You can do anything you want (except divide by zero) toone side of an equation, as long as you do the same thing to theother.


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5 Is there more than one mathematical operation acting on"d"?

Yes

No

dts =

Rule 3. If there is more than one operation going on, you must undothem in the opposite order in which you would do them, the oppositeof the "order of operations."


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The order of the s and t doesn'tmatter, but we usually make them

alphabetical so they look better.

Are we done?

Applying Rules 1 and 2

dt

s =


2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same thing to the other.So we undo d being divided by t, by multiplying both sides by t.

dt

s =(t)(t)

ds =t

dst =


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We've now solved our equation for d.

A harder problem is to solve it for t.

Applying Rule 4

Rule 4. You can always switch the left and right sides of an equation.

dst =

d = st


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Let's solve this equation for "t"

Solving for t

dt

s =

That means that when we're donewe'll have t alone



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6 Is t already ALONE? If not, what is with it?

A s

B d

C t


dt

s =


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7 What mathematical operation connects d to t?

A t is being divided by d

B d is being divided by t

C d is being multiplied by t

D t is being subtracted from d

dt

s =


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8 What is the opposite of dividing d by t?

A dividing d by t

B dividing s by t

C multiplying d by t

D multiplying t by d

dt

s =

Rule 1. To "undo" a mathematical operation, you must do theopposite.


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9 What must we do if we multiply the right side by t?

A divide the left side by t

B multiply the left side by t

C divide the left side by dD divide the left side by d

dt

s =

Rule 2. You can do anything you want (except divide by zero) toone side of an equation, as long as you do the same thing to theother.


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10 Is there more than one mathematical operation acting on"d"?

Yes

No

dt

s =

Rule 3. If there is more than one operation going on, you must undothem in the opposite order in which you would do them, the oppositeof the "order of operations."


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Are we done?

Solving for T

dt

s =


2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same thing to the other.So we undo d being divided by t, by multiplying both sides by t.

dt

s =(t)(t)

dst =


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11 Is t ALONE? If not, what is with it?

A s

B d

C t


st = d


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12 What mathematical operation connects s to t?

A t is being divided by d

B t is being divided into s

C t is being multiplied by s

D t is being subtracted from s

st = d


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13 What is the opposite of multiplying t by s?

A dividing t by s

B dividing t by t

C multiplying t by t

D multiplying by t by s

st = d


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d=s

sts

st = d

s

ds

t =

Solving for t

2. You can do anything you want (except divide by zero) to oneside of an equation, as long as you do the same thing to theother.


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14 Is t ALONE on the left?

A s

B d

C t

D it is alone

ds

t =


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Let's solve this equation for "vo "

Solving for vo

v = vo + at

That means that when we're done

we'll have vo alone



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15 Is vo already ALONE? If not, what is with it?

A only a

B only t

C at


v = vo + at


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16 What mathematical operation connects at to v o?

A "at" is being divided by vo

B "at" is being added to vo

C vo is being multiplied by "at"

D vo is being divided by "at"

v = vo + at


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17 What is the opposite of adding at to vo ?

A dividing by vo by "at" into t

B subtracting v o from "at"

C subtracting "at" from vo

D dividing "at" by vo

v = vo + at


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18 What must we do, if we subtract "at" from the right side?

A add "at" to the left side

B multiply the left side by "at"

C subtract "at" from the left side

D divide the left side by v o

v = vo + at


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19 Is there more than one mathematical operation acting on

"vo "?

Yes

No

v = vo + at


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2. You can do anything you want (except divide by zero) to oneside of an equation, as long as you do the same thing to the

other.

Solving for vo

v = vo + at- at - at

v - at = vo

vo = v - at


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Let's solve this equation for "a"

Solving for a

v = vo + at

That means that when we're donewe'll have a alone



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20Is a already ALONE? If not, what is with it?

A only vo

B only t

C vo and t


v = vo + at


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21 What mathematical operation connects vo to at?

A "at" is being divided by vo

B vo is being added to "at"

C vo is being multiplied by "at"

D vo is being subtracted by "at"

v = vo + at


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22What is the opposite of adding vo to at?

A dividing by vo by at into t

B subtracting vo from at

C subtracting at from vo

D dividing at by vo

v = vo + at


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23 What mathematical operation connects t to a?

v = vo + at

A a is added to t

B a is multiplied by t

C a is divided by t

D t is subtracted from a


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24 What is the opposite of multiplying a by t?

v = vo + at

A dividing a by t.

B dividing t by a.

C multiplying a by t

D multiplying t by a


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25 What must we do, if we divide by t from the right side?

A divide the left side by a

B multiply the left side by a

C divide the left side by t

D multiply the left side by t

v = vo + at


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26 Is there more than one mathematical operation acting on"a"?

Yes

No

v = vo + at


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27 Which operation should we undo first?

A divide a by t

Bsubtract v

o

from at

v = vo + at


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28 Which operation should we undo second?

A divide a by t

Bsubtract v

o

from at

v = vo + at


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Are we done?

1. To "undo" a mathematicaloperation, you must do the opposite.

2. You can do anything you want

(except divide by zero) to one side ofan equation, as long as you do thesame thing to the other.

3. If there is more than one operationgoing on, you must undo them in theopposite order in which you would do

them, the opposite of the "order ofoperations."

Solving for a

v = vo + at- vo - vo

v - vo = at

tt

v - vo = a

t


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Let's solve this equation for "t"

Solving for t

v = vo + at

That means that when we're donewe'll have t alone



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1. To "undo" a mathematical operation, youmust do the opposite.

2. You can do anything you want (exceptdivide by zero) to one side of an equation, aslong as you do the same thing to the other.

3. If there is more than one operation goingon, you must undo them in the opposite orderin which you would do them, the opposite ofthe "order of operations."

Solving for t

v = vo + at

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