There are two prices (for members and non-members) for tickets to the Houston Zoo Lights Christmas festival. If 5 members and 12 non-members attend the event it will cost $160.

Question

If 2 members and 5 non-members attend it will cost $66. Determine the price of a non-member ticket.

Answer 1

The first thing we will want to do is make some equations based on the information we have. Our variables (the information we don't know) are the prices of tickets. We do know there are two types of tickets with different prices (member tickets and non-member tickets).

Let's use x to represent the price of member tickets and y to represent the price of non-member tickets.

x= member ticket y=non-member ticket

5 member tickets and 12 non-member tickets costs $160. We can translate that into an equation now that we have specified what our variables are and what each letter represents.

5x + 12y = 160

We also know that 2 member tickets and 5 non-member tickets costs $66. Using the same variables as before we can make another equation.

2x + 5y = 66

The next step is to combine these equations and cancel out one of our variables so we can solve for the other. Just looking at these equations I can see that simply adding or subtracting them is not going to cancel out either our x or y. But do not fear!

We are going to multiple each equation by a number. It looks like our x's are going to be easier to cancel each other out. What are common multiples of our x coefficients (5 and 2, in this problem)? They are both divisible by 10, 20, 30, etc. 10 is the smallest number so let's choose that.

We want to get each of our x coefficients (the number of member tickets) to equal ten.

Let's take a look again at our first equation: 5x + 12y = 160

What can we multiply to 5 to get 10? 2!

So we multiply the entire equation by 2. Because we are distributing this 2 to every part of the equation it does not change it's value.

(5x + 12y = 160)*2

(5x)*2 + (12y)*2 = (160)*2

10x + 24y = 320

Now let's do the same for our second equation: 2x + 5y = 66

We want to be able to combine these equations so that all our x's cancel out. So this time we want to end up with a negative 10.

2 times what equals -10? -5

Distribute the -5, pay close attention to the negative

(2x + 5y = 66)*(-5)

(2x)*(-5) + (5y)*(-5) = (66)*(-5)

-10x - 25y = -330

And now time to add our equations!

10x + 24y = 320

+

-10x - 25y = -330

=

-y = -10

Our x's cancel out. Now we just have to divide or multiply by negative 1 (either will work) to get a positive value for y.

y = 10

Now what did y stand for again? The price of the non-member ticket.

And what was the question asking for originally? The price of a non-member ticket!

And what were those units again? Dollars!

y = $10; a non-member ticket costs $10

There is our answer!

CHECKING OUR WORK and finding x

But it's not over yet!

Simply plug in our y value into either equation to find the value of x.

5x + 12*(10) = 160

5x + 120 = 160

subtract 120 from each side of the equals sign

5x = 40

divide by 5

x= 8; a member ticket costs $8

From here you can plug in both those values to the original equations to confirm that everything is right!

NOTE: when combining equations you can add or subtract. I chose to multiply our second equation by a negative number and add the equations. You could also have multiplied the second equation by a positive 5 and then subtracted the two equations. Whatever is easiest for you!