I am struggling with my Systems of Equations

Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

The box office at a theater is selling tickets for a series of rock concerts. So far, they have sold 98 balcony tickets and 70 general admission floor tickets for Friday's show, for a total of $6,888 in receipts. For Saturday's show, 85 balcony tickets and 30 general admission floor tickets have been sold, equaling $5,145 in receipts. How much does each ticket cost?

Answer 1

There are three kinds of tickets: Balcony and General admission. Let's call the price of Balcony tickets, B, and the price of General admission tickets, G.

"They have sold 98 balcony tickets and 70 general admission floor tickets for Friday's show, for a total of $6,888"

98 balcony tickets cost 98*B, and 70 general admission tickets cost 70*G. Total cost is 98*B+70*G and is equal to $6888. So, out first equation is:

98*B + 70*G = 6888

If we use the same strategy, the second equation is

85*B + 30*G = 5145

Let's solve this:

98*B + 70*G = 6888

85*B + 30*G = 5145

We have 70*G and 30*G. If we multiply first equation with 3 and second with 7, we get

3*98*B + 210*G = 3*6888

7*85*B + 210*G = 7*5145

Subtract second equation from the first one. The G terms will cancel:

(3*98 - 7*85)*B = 3*6888 - 7*5145

-301 * B = -15351

B = 51

Let's substitute B=51 to the first equation

98*51 + 70*G = 6888

4998 + 70*G = 6888

70*G = 1890

G = 27

Answer 2

The idea of systems of equations is to find two equations, and then find at what point do those equations intersect. There are multiple ways to do it, but probably the easiest method in this particular problem would be the elimination method (also sometimes called the addition method)

From the problem given, we know that there are two types of tickets, balcony and general admission. We know that we sold 98 balcony tickets and 70 general tickets, for a total of 6,888. the problem is that we don't know how much the tickets cost, so we will let:

b = cost of balcony tickets

g = cost of general ticket.

So the first equation would be 98b + 70g = 6,888

The idea being that we would plug in the prices for each and they should equal 6,888, per what is stated in the problem.

For the second equation, we know that we sold 85 balcony tickets and 30 general tickets. So this equation would be

85b + 30g = 5,145

Now lets do the elimination method. The idea here is to pick one variable from both equations and try to get them to be their least common multiple through multiplication. However one of them needs to be negative and the other needs to be positive. So in our two equations we have

98b + 70g = 6,888

85b + 30g = 5,145

Just glancing at these, it would be hard to get the b term to be the same because we are dealing with 98 and 85, and they are unlikely to have an easy least common multiple. However, with the g term, we have 70 and 30. The least common multiple of these two numbers is 210. So what we do is multiply the ENTIRE top equation by 3 and multiply the ENTIRE bottom equation by -7. We multiply the bottom by negative 7 because, as stated earlier, we want one of them to be negative and the other to be positive. now lets do the multiplication

3 (98b + 70g = 6,888) = 294b + 210g = 20,664 = This is our new first equation!

-7 (85b + 30g = 5,145) = -595b -210g = -36015 = This is our new second equation!

Now, what we do is ADD the two new equations by combining like terms.

294b + -595b = -301b

210g + -210g = 0

20,664 + -36,015 = -15,351

So our new added equation is -301b + 0 = -15,351

This simplifies to -301b = -15,351

Now our mission is to get b by itself. Since it is currently being multiplied by -301, we need to divide both sides by -301. If we do that we get

b = (-15,351 / -301) = 51 = b = cost of a balcony ticket

Now that we know what be is, we can plug it back into either of the original equations to find what the cost of a general ticket is. I'll plug it into the first equation we found

98 (51) + 70g = 6,888

4998 + 70g = 6,888

Now we are trying to get g by itself so we will subtract 4,998 from both sides of the equation THEN divide 70 on both sides of the equation

When we subtract 4998 we get 70g = 6,888 - 4998 -------->>> 70g = 1890 ----------->>> g = 27

So now we know that the cost of the balcony ticket is $51 and the cost of a general ticket is $27.