Tickets for the school play cost $$ \$ 10 $$ for adults and $$ \$ 5 $$ for students. 120 tickets were sold in all for $$ \$ 800 $$. The equations for total profit and total tickets are as follows: $$ 5 x+10 y=800 $$ and $$ x+y=120 $$ tickets, where $$ x $$ is the number of students and $$ y $$ is the numt of adults.

Question

UpStudy AI · Accepted Answer

To determine the number of student and adult tickets sold for the school play, we'll solve the system of equations provided:

$$
\begin{cases}
5x + 10y = 800 \quad \text{(Total Revenue)} \
x + y = 120 \quad \text{(Total Tickets Sold)}
\end{cases}
$$

Where:
- $$ x $$ = Number of student tickets sold
- $$ y $$ = Number of adult tickets sold

### Step 1: Simplify the Equations

First, let's simplify the revenue equation by dividing all terms by 5 to make the calculations easier:

$$
\frac{5x + 10y}{5} = \frac{800}{5} \
x + 2y = 160 \quad \text{(Simplified Revenue Equation)}
$$

Now, our system of equations looks like this:

$$
\begin{cases}
x + 2y = 160 \quad \text{(1)} \
x + y = 120 \quad \text{(2)}
\end{cases}
$$

### Step 2: Subtract Equation (2) from Equation (1)

Subtracting equation (2) from equation (1) helps eliminate $$ x $$:

$$
(x + 2y) - (x + y) = 160 - 120 \
x + 2y - x - y = 40 \
y = 40
$$

### Step 3: Substitute $$ y $$ Back into Equation (2)

Now that we know $$ y = 40 $$, substitute this value back into equation (2) to find $$ x $$:

$$
x + 40 = 120 \
x = 120 - 40 \
x = 80
$$

### Step 4: Conclusion

- **Number of Student Tickets Sold ($$ x $$)**: **80**
- **Number of Adult Tickets Sold ($$ y $$)**: **40**

### Verification

To ensure the solution is correct, let's verify it against the original revenue equation:

$$
5x + 10y = 5(80) + 10(40) = 400 + 400 = 800
$$

The total revenue matches the given amount of \$800, confirming that the solution is correct.

### Summary

- **Student Tickets Sold**: 80
- **Adult Tickets Sold**: 40
There were 80 student tickets and 40 adult tickets sold.

Tickets for the school play cost for adults and for students. 120 tickets were sold in all for
. The equations for total profit and total tickets are as follows:
and tickets, where is the number of students and is the numt
of adults.

Upstudy AI Solution

Answer

Solution

Step 1: Simplify the Equations

Step 2: Subtract Equation (2) from Equation (1)

Step 3: Substitute Back into Equation (2)

Step 4: Conclusion

Verification

Summary

Bonus Knowledge

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Tickets for the school play cost for adults and for students. 120 tickets were sold in all for . The equations for total profit and total tickets are as follows: and tickets, where is the number of students and is the numt of adults.