Calculate X and Y: How to Determine Special Price Tickets for a Group of 14

[Warz]

Hi,

How do you calculate x y?

Lets make an example to show what I mean:
Say there are 14 people who are buying tickets for a buss trip. In total all tickets costs 2040$. So, we got 14 people paying total 2040$. An ordinary ticket costs 160$ but there is a special price for X number of these people and their ticket will cost only 120$.

So its 160x + 120y = 2040 and x+y=14, right?

I don't get it.. How to I calculate this now? I'm trying to figure out how many has special price ticket. Any ideas?

Thanks

 

[danago]

Have you learned any methods of solving systems of simultaneous equations?

What happens if you multiply everything in the second equation by 120 and then subtract it from the first equation?

 

[arildno]

Well, you can work a bit on the first equation:

IF 160x+120y=2040, then obviously, we must have:

40x+120x+120y=2040. Agreed?

Now, since 120x+120y can be written as 120x+120y=120*(x+y), IRRESPECTIVE of the values x and y must have, we may rewrite our first equation as:

40x+120*(x+y)=2040 (*)

Now, can you simplify (*) further by means of your second equation, namely x+y=14?

 

[Warz]

Wow you are really smart, didn't think about it that way!
I got x = 9 now by doing 40x+120*14 = 2040

it seems to be very right! thank you :)

 

[arildno]

Before leaving, think about the proposal of Danago.

He said first:

1. Multiply your second equation with 120 (ask yourself, hmm..why?)
IF we do that we have:

120*(x+y)=120*14
or, simplified:
120x+120y=1680.

2. Now, since 120x+120y is the same number as 1680 (that's what your equation SAYS!), clearly, we can subtract 120x+120y from the left-hand side of equation 1, whereas we subtract 1680 from the right-hand side:

160x+120y-(120x+120y)=2040-1680
Simplifying this yields:
40x=360
which you can readily solve for x=9

Danago's approach is more general than the one I proposed, but it seems a bit "strange" at first.

 

[Warz]

I see.. now I'm trying with something more difficult: 98x+43y=2942 and x+y=14

Now? I say
x*98 + y*98 = 14*98

43y=2942-1372
43y=1570 |/43
y=1570/43

right?

then we can put y into 98x+43(1570/43)=2942
98x=2942-1570
98x=1372
x=14

Now let's see if correct (1570/43) + 14 = 50.51162791 :/ Wrong... Am i thinking wrong now?

 

[CRGreathouse]

I see.. now I'm trying with something more difficult: 98x+43y=2942 and x+y=14

Now? I say
x*98 + y*98 = 14*98

43y=2942-1372
43y=1570 |/43
y=1570/43

right?

98x + 98y = 14*98 = 1372 is good. But this gives you
(98x+43y) - (98x + 98y) = 2942 - 1372 = 1570
(43 - 98)y = -55y = 1570

not
43y=2942-1372.