
Sarah took the advertising department from her company on a round trip to meet with a potential client. Including Sarah a total of 8 people took the trip. She was able to purchase coach tickets for ​$140 and first class tickets for ​$1050. She used her total budget for airfare for the​ trip, which was ​$5670. How many first class tickets did she​ buy? How many coach tickets did she​ buy?

Let's define variables to represent the number of coach and first-class tickets that Sarah bought:

- Let $ x $ be the number of first-class tickets.
- Let $ y $ be the number of coach tickets.

We know the following:

1. The total number of people is 8, so:
   \[
   x + y = 8
   \]
   
2. The total budget for airfare is $5670, and the cost of a first-class ticket is $1050 while the cost of a coach ticket is $140. So the total cost of the tickets is:
   \[
   1050x + 140y = 5670
   \]

Now we have a system of two equations:

1. $ x + y = 8 $
2. $ 1050x + 140y = 5670 $

### Step 1: Solve the first equation for $ y $:
\[
y = 8 - x
\]

### Step 2: Substitute $ y = 8 - x $ into the second equation:
\[
1050x + 140(8 - x) = 5670
\]
Simplify the equation:
\[
1050x + 1120 - 140x = 5670
\]
\[
910x + 1120 = 5670
\]
Subtract 1120 from both sides:
\[
910x = 4550
\]
### Step 3: Solve for $ x $:
\[
x = \frac{4550}{910} = 5
\]

So, Sarah bought 5 first-class tickets.

### Step 4: Find $ y $ (the number of coach tickets):
\[
y = 8 - x = 8 - 5 = 3
\]

Thus, Sarah bought **5 first-class tickets** and **3 coach tickets**.

Would you like more details or have any further questions?

Here are some related questions:

1. What is the difference in total cost if she had bought 4 first-class tickets instead of 5?
2. How much would she save if she bought 8 coach tickets instead of a combination?
3. What would the total cost be if the price of a coach ticket increased by 10%?
4. How many people could she take if her budget was increased to $7000, but the prices of the tickets stayed the same?
5. If the client covered the cost of one first-class ticket, how would that affect the number of tickets Sarah could afford?

**Tip:** In systems of equations, substitution is often useful when you can easily isolate one variable from an equation.

Sarah took the advertising department from her company on a round trip to meet with a potential client. Including Sarah a total of 8 people took the trip. She was able to purchase coach tickets for ​$140 and first class tickets for ​$1050. She used her total budget for airfare for the​ trip, which was ​$5670. How many first class tickets did she​ buy? How many coach tickets did she​ buy?

Learn how to solve a system of equations to determine how many first-class and coach tickets were bought within a $5670 budget. The solution covers key algebraic concepts like linear equations and substitution methods. Suitable for students in Grades 8-10.

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