Martin’s Hardware is clearing out their inventory of bicycles and tricycles. The clerk rolls all of the bicycles and tricycles outside and arranges them in a line on the sidewalk. If the total number of bicycles and tricycles is 37 and the total number of wheels on all the bicycles and tricycles combined is 90, how many tricycles are in the line on the sidewalk?
For this question, wouldn't you use linear programming? If so, can someone explain and help me with it? I really don't understand.

Question

Martin’s Hardware is clearing out their inventory of bicycles and tricycles.  The clerk rolls all of the bicycles and tricycles outside and arranges them in a line on the sidewalk.  If the total number of bicycles and tricycles is 37 and the total number of wheels on all the bicycles and tricycles combined is 90, how many tricycles are in the line on the sidewalk?
For this question, wouldn't you use linear programming? If so, can someone explain and help me with it? I really don't understand.

phi · Answer

I think this is just ordinary algebra.

The first step is assign variable names to the "unknowns"

anonymous · Answer

Let b and t be the number of bicycles and tricycles respectivley.
Solve the following equations for b and t:$$\{b+t=37,2b+3t=90\}$$$$\{b=21,t=16\} $$

phi · Answer

they ask for
how many tricycles

so T  for # of tricycles  would be a good name for a variable


they say
 total number of bicycles and tricycles is 37 

they mention the number of bicycles  and tricycles. 
so B for # of bicycles would be good.

translate the sentence
total number of bicycles and tricycles is 37 

into math
B + T = 37

that is short for "# of bikes plus # of trikes is 37

phi · Answer

you also have to translate
 total number of wheels on all the bicycles and tricycles combined is 90

into math.

tobey already posted the equations, but if you want to figure it out yourself,
you have to think"  how many wheels does a bike have?"
and then " if I have B bikes  (B is the (unknown) # of bikes) )  how many wheels is that? "   you should think:  2 wheels per bike times B bikes:  2B wheels

use the same idea for the number of wheels on a trike to get  3T wheels
add them up to get

2B+3T = 90

phi · Answer

once you have the equations
(remember:  for every unknown you need an equation... in this case you have two unknowns, B and T, and 2 equations)  so you can solve for both variables.

B + T = 37
2B+3T = 90

can you solve these?

anonymous · Answer

Thank you!!

phi · Answer

just to finish,   you can solve using elimination or substitution

elimination:  multiply the first equation by -2  (that means both sides, all terms)
to get
-2B + -2T = -2*37
add this equation to the second equation
  2B+3T = 90
to get

-2B + 2B + -2T + 3T = -2*37 + 90
and simplify

phi · Answer

-2B + 2B is 0  and -2T + 3T is 1T, so the left side is
1T = -2*37 + 90
or
T =  -74 + 90  
or
T= 90 - 74 = 16