Word Problems





Watch the below videos for slow lessons on the Elimination Method

Elimination Method Video-1 


Elimination Method Video-2    


now for your homework answers


                 Writing Systems to Model Situations
                -------------------------------------

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?




       Label your variables first!!!!


       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  



================================================================================






                                   Restaurant Menu Problems
                                _______________________________


  
 
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.



First ==>>  label your variables


             C  #  chicken burritos    S  # steak burritos


Second express your two equations


                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 
your answer.



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


      Second express your equations


                      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