Question

question 13 (essay worth 10 points)
(real world systems of equations hc)
visitors to a state fair can buy an unlimited - ride pass for $48 and an entrance - only pass for $26. in one day, 264 passes were sold for a total of $10,560.
part a: define the variables and write a system of equations that could be used to determine the number of each type of pass sold (2 points)
part b: determine how many unlimited - ride passes and entrance - only passes were sold. show all work (4 points)
part c: explain how you know your answer in part b is correct. show any needed work. (4 points)

Explanation:

Part A:

Step1: Define variables

Let $x$ = number of unlimited-ride passes sold, $y$ = number of entrance-only passes sold.

Step2: Total passes equation

Total passes sold: $x + y = 264$

Step3: Total revenue equation

Total revenue: $48x + 26y = 10560$

Part B:

Step1: Isolate $y$ from first eq

$y = 264 - x$

Step2: Substitute into second eq

$48x + 26(264 - x) = 10560$

Step3: Simplify and solve for $x$

$48x + 6864 - 26x = 10560$
$22x = 10560 - 6864$
$22x = 3696$
$x = \frac{3696}{22} = 168$

Step4: Solve for $y$

$y = 264 - 168 = 96$

Part C:

Step1: Verify total passes

$168 + 96 = 264$, matches given total.

Step2: Verify total revenue

$48(168) + 26(96) = 8064 + 2496 = 10560$, matches given revenue.

Answer:

Part A:

Variables: $x$ = unlimited-ride passes sold, $y$ = entrance-only passes sold
System of equations:

$$\begin{cases} x + y = 264 \\ 48x + 26y = 10560 \end{cases}$$

Part B:

168 unlimited-ride passes and 96 entrance-only passes were sold.

Part C:

The answer is correct because substituting the values back into both original equations satisfies the total number of passes sold ($168+96=264$) and the total revenue ($48(168)+26(96)=10560$), which matches the given values.