# Word Problems

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Watch the below videos for slow lessons on the Elimination Method

Elimination Method Video-1

Elimination Method Video-2

Writing Systems to Model Situations
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1: The admission fee at a carnival is \$3.00 for children and \$5.00 for adults. On the first day, 1,500 people entered the fair and \$5,740 was collected.
How many children and adults went to the carnival?

Pick meaningful variable names  :

A for number of adults who went

C for number of children who went

what you know so far....

3C  +  5A  =  5740   what the total amount of tickets cost

C  +   A  =  1500   number of children and adults who went

now let's solve it....

I want to get rid of Cs first, I could get rid of As first, but in this case, I picked Cs

I got 3C in the first row, so if I multiply the second row by -3...to make sure I create a  -3C so   the Cs in both rows add to  0

so I did  -3( C  +  A =  1500 ) to get  -3c -3C =  -4500

3C  +  5A  =  5740
-3C  -  3A  = -4500
-------------------------------------

2A  =  1240

A  =   620

now plug A back into the equation  A + C  =  1500

A  +  C  = 1500

620 +  C  =  1500
-620          -620
------------------------

C  =  880

So ( adults , children )  is  ( 620 , 880 )

2:  A builder placed two orders with a hardware store.

a:  The first order was for 25 sheets of plywood and 4 boxes of nails and the total bill was  \$357.

b:  The second order was for 35 sheets of plywood and 2 boxes of nails and the bill totaled \$471.

What were the prices of one piece of plywood and one box of nails?

N  # of nails

P  # of pieces of plywood

25P  +  4B  =  357

35P  +  2B  =  471

removing the Bs first is easier, so lets create -4b to get rid of +4b

-2( 35P +  2B  =  471 ) is  -70P - 4B = -942

25P +  4B  =  357
-70P -  4B  = -942
--------------------------

-45P       =  -585

-45P  =  -585
-----    ------
-45      -45

P  =  13

plug P into the top equation

25P + 4B = 357

25(13) + 4B  =  357

325 + 4B =  357
-325        -325
--------------------
4B =    32

B  = 8

3:  Two friends bought some markers and pens.

M # of markers

P # of pens

a:    The first bought 4 markers and 5 pens and it cost him \$6.71.

4M +  5P  =  6.71

b:     the second friend bought 5 markers and 3 pens, which cost her \$7.12.

5M  +  3P  =  7.12

these are our two equations

question is, do we get rid of M or P first?....actually doesn't matter

let's get rid of P first

-3( 4M + 5P =  6.71 )
5( 5M + 3P =  7.12 )

-12M - 15P =  -20.13
25M + 15P =   35.60
----------------------------
13M       =   15.47

M =  1.19

plug M into either equation to find P

5(1.19) +  3P = 7.12

5.95 +  3P  =  7.12
-5.95        =  -5.95
-----------------------
3P  =   1.17

P  =  0.39

Pens = 0.39  Markers = 1.19   (  0.39 , 1.19 )

4.  The ticket price for the movies is \$7.50 for children and \$10.50 for adults.

One night 825 people bought tickets and \$8005.50 was collected from tickets sales.  How many children and adults bought tickets?

Label   A  for # of adults   and  C for # of children

a:            A  +      C  =    825
10.50A  +  7.50C  =   8005.50
-----------------------------------

this is going to get real messy...so don't get concerned about it

-7.50(  A  +  C  =  825  )  I want to get rid of the C first

-7.50A  -  7.50C   =   -6187.50

-7.50A   -  7.50C   =  -6187.50
10.50A   +  7.50C   =   8005.50
-----------------------------------
3A                =   1818

A =  606

now plug P back into the first equation

A +  C  = 825

606  +  C  =   825
-606           -606
-----------------------

C  =   219

Adults = 606   Children = 219

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5:  The Smith family orders two chicken burritos and three steak burritos. These contain  3460 calories. The Grant family orders three chicken burritos
and two steak burritos. These contain 3340 calories. Find the number of calories in each chicken burrito and in each steak burrito. Justify your answer.

C  #  chicken burritos    S  # steak burritos

2C +  3S  =  3460
3C +  2S  =  3340

do I get rid of C or S first..?..doesn't matter..either choice works

so I pick C..... but how..?   2C +  3C don't add to 0, I need them to add to 0 to get rid of C

so I pull this trick   -3( 2C ) + 2( 3C ) =   -6C + 6C = 0

so I multiply the top equation by -3 and the bottom equation by 2

-3( 2C + 3S = 3460 )
2( 3C + 2S = 3340 )

which turns into  -6C - 9S  =  -10380
+6C + 4S  =    6680
----------------------
-5S =  -3700
----   -----
-5       -5

S = 740

now plug S=740 back into one of the original equations, doesn't matter which one

2C + 3S     =  3460

2C + 3(740) =  3460

2C + 2220   =  3460
- 2220     -2220
---------------------
2C          =  1240
----           ------
2                2

C  =  620

6:  The Smith family orders four sides of chips and four sides of beans. These contain 1980 calories. The Grant family orders two sides of chips and
five sides of beans. These contain 1350 calories. Find the number of calories
in each side of ships and number of calories in each side of beans. Justify

First label your variables  ==>>  C for chips   B for beans

4C + 4B  =  1980
2C + 5B  =  1350

lets get rid of  C first

create -4C from the bottom equation..multiply it by -2

4C + 4B  =  1980
-2( 2C + 5B  =  1350 )

which becomes

4C + 4B  =  1980
-4C -10B  = -2700
-------------------------

-6B  =  -720
-----   ------
-6       -6

B = 120

plug B back into the top equation to get C

4C  +  4B    =  1980

4C  + 4(120) =  1980

4C  +   480  =  1980
-   480  =  -480
-------------------------
4C           =  1500

4C  =  1500
----   ------
4      4

C = 375

7:  The Smith family orders three iced teas and three juice drinks which contain 1260 calories.

Label variables first  T : ice tea     J : juice

3T  +  3J  =  1260 ( 1st equation )

The Grant family orders one ice tea and 4 juice drinks, total calories of 1140.

1T  +  4J  =  1140 ( 2nd equation )

let's get rid of T first, so multiply the 2nd equation
by -3 to create  -3T

3T  +  3J  =  1260   ==>>    3T + 3J  =   1260
-3( 1T  +  4J  =  1140 ) ==>>   -3T -12J  =  -3420
---------------------------       -------------------
-9J   =  -2160
----    -------
-9         -9

J =  240
3T +  3J     =  1260
3T +  3(240) =  1260
3T +  720    =  1260
-  720       -720
---------------------
3T           =   540
----             -----
3                 3

T = 180

plug J=240 back in the first equation

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