Fiona's Fashion Store is world renowned for its buttoned uniforms. A collection of 36 shirts and 42 jackets contains 842 buttons. A collection of 6 shirts and 7 jackets contains 137 buttons. Each shirt has the same number of buttons, and each jacket has the same number of buttons How many buttons are there in a Fiona's Fashion shirt, and how many buttons are there in a jacket?

Let x represent the number of buttons in a shirt and let y represent the number of buttons in a jacket. Since we have two unknowns, we need two equations to find them.

Let's use the given information in order to write two equations containing x and y. For instance, we are given that a collection of 36 shirts and 42 jackets contains 842 buttons. How can we model this sentence algebraically?

The total buttons contained in shirts can be modeled by 36x36x36, x, and the total buttons contained in jackets can be modeled by 42y42y42, y. Since these together add up to 842842842, we get the following equation:

36x+42y = 84236x+42y=84236, x, plus, 42, y, equals, 842

We are also given that a collection of \textit{6}66 shirts and \textit{7}77 jackets contains \textit{137}137137 buttons. This can be expressed as:

6x+7y=1376x+7y=1376, x, plus, 7, y, equals, 137

Now that we have a system of two equations, we can go ahead and solve it!

Hint #33 / 4

We can now solve the system of equations by the elimination method. Note that the coefficient of y in the first equation, 42, is exactly 6 times the coefficient of y in the second equation, 7. Therefore, we can multiply the second equation by \purple -6 start color purple, minus, 6, end color purple in order to eliminate y.

-6 6x+-6) 7y{-6}36x-42y&=-822

−6⋅6x+(−6)⋅7y

−36x−42y

=−6⋅137

=−822

Now we can eliminate y:

36x+42y+ −36x−42y0=842=−822=20

Since 0 200≠200, does not equal, 20, we got a false statement, which means that the system has no solutions.

Because the system has no solutions, there is no definite number of buttons in a shirt and a jacket as described in the question. In other words, the given information describes an impossible situation.