There are a total of 135 seats, 118 front handlebars (that steer the bike), and 269 wheels. How many bicycles, tandem bicycles and tricycles are there in store?

Question

Not sure how to start how or how to solve. Have to be able to show work.

Answer 1

x represents number of bicycles

y represents number of tandems

z represents number of tricycles

 

From the seats

x + 2y + z = 135

 

From the handle bars

x + y + z = 118

 

From the wheels

2x + 2y + 3z = 269

 

Subtract the "bars" from the "seats."

y = 17

 

Multiply both sides of "bars" by 2 and subtract from "wheels."

z = 33

 

x + 17 + 33 = 118

x + 50 = 118

x = 68

 

check:

68 + 34 + 33

135

 

136 + 34 + 99

269

 

 

 

Answer 2

Hi Liy,

 

Let's begin by assigning three letters to represent our three unknowns:

    x = number of bikes

    y = number of tandems

    z = number of tricycles

 

Now let's come up with a system of three equations to solve for our three unknowns:

 

    Equation #1 (based upon number of seats):

        x + 2y + z = 135 (total seats)

                              x (1 seat per bike)

                              y (2 seats per tandem)

                              z (1 seat per tricycle)

 

    Equation #2 (based upon number of handle bars)

        x + y + z = 118 (total handle bars)

                             x (1 handle bar per bike)

                             y (1 handle bar per tandem)

                             z (1 handle bar per tricycle)

 

    Equation #3 (based upon number of wheels)

        2x + 2y + 3z = 269 (total wheels)

                                 x (2 wheels per bike)

                                 y (2 wheels per tandem)

                                 z (3 wheels per tricycle)

 

Now let's group our three equations together for solution and manipulate them to solve for each unknown:

 

    Equation #1:  x + 2y + z = 135

    Equation #2:  x + y + z = 118

    Equation #3:  2x + 2y + 3z = 269 

 

    Equation #4:  Equation #1 - Equation #2

                        (eliminate unknowns x and z)

    Equation #4:  y = 17 (number of tandems)

 

    Equation #5:  (replace y in Equation #2

                         with 17 from Equation #4)

    Equation #5:  x + 17 + z = 118

    Equation #5:  x + z = 118 - 17

    Equation #5:  x + z = 101

 

    Equation #6:  (replace y in Equation #3

                         with 17 from Equation #4)

    Equation #6:  2x + 2(17) + 3z = 269

    Equation #6:  2x + 34 + 3z = 269

    Equation #6:  2x + 3z = 269 - 34

    Equation #6:  2x + 3z = 235

 

    Equation #7:  (multiply each term in Equation #5

                         by 2)

    Equation #7:  2x + 2z = 202

 

    Equation #8:  Equation #6 - Equation #7
                        (eliminate unknown x)
    Equation #8:  z = 33 (number of tricycles)

 

    Equation #9:  (replace y in Equation #2

                        with 17 from Equation #4 and

                        replace z in Equation #2

                        with 33 from Equation #8)

    Equation #9:  x + 17 + 33 = 118

    Equation #9:  x + 50 = 118

    Equation #9:  x = 118 - 50

    Equation #9:  x = 68 (number of bikes)

 

Finally, we can verify each of our answers by plugging them back into our three original equations and evaluate each equation to ensure that each equation is true:

 

    Equation #1:  x + 2y + z = 135

                        68 + 2(17) + 33 = 135

                        68 + 34 + 33 = 135

                        102 + 33 = 135

                        135 = 135 (equation is true)

 

    Equation #2:  x + y + z = 118

                        68 + 17 + 33 = 118

                        85 + 33 = 118

                        118 = 118 (equation is true)

 

    Equation #3:  2x + 2y + 3z = 269

                        2(68) + 2(17) + 3(33) = 269

                        136 + 34 + 99 = 269

                        170 + 99 = 269

                        269 = 269 (equation is true)

 

Since we see that all three original equations evaluated as true using each of our answers, we are confident that our answers are correct.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.