Let's tackle each question step by step, starting with Question 7:
### Question 7
We need to determine the number of solutions for the given system of equations:
[tex]\[
\begin{cases}
-2x - 6y = -20 \\
x + 3y = 11
\end{cases}
\][/tex]
To solve this, let's analyze the equations:
1. First equation: [tex]\(-2x - 6y = -20\)[/tex]
2. Second equation: [tex]\(x + 3y = 11\)[/tex]
First, let's multiply the second equation by 2 to facilitate the elimination method:
[tex]\[
2(x + 3y) = 2 \cdot 11 \implies 2x + 6y = 22
\][/tex]
So, the modified system is:
[tex]\[
\begin{cases}
-2x - 6y = -20 \\
2x + 6y = 22
\end{cases}
\][/tex]
Now, if we add these two equations:
[tex]\[
(-2x - 6y) + (2x + 6y) = -20 + 22 \implies 0 = 2
\][/tex]
This is a contradiction, indicating that the system of equations has no solutions. Therefore, the answer is:
(D) No solutions
### Question 8
We need to find the number of adult and kid tickets sold given the total ticket sales and total attendance:
1. Each adult ticket costs \[tex]$20
2. Each kid ticket costs \$[/tex]12
3. Total attendance is 840
4. Total ticket sales amount to \$12,496
Let's define:
- [tex]\(a\)[/tex] as the number of adult tickets sold
- [tex]\(k\)[/tex] as the number of kid tickets sold
The two equations we can set up are:
1. [tex]\(a + k = 840\)[/tex] (total attendance)
2. [tex]\(20a + 12k = 12,496\)[/tex] (total ticket sales)
We will solve these equations step by step.
From the first equation, solve for [tex]\(a\)[/tex]:
[tex]\[
a = 840 - k
\][/tex]
Substitute [tex]\(a = 840 - k\)[/tex] into the second equation:
[tex]\[
20(840 - k) + 12k = 12,496
\][/tex]
[tex]\[
16,800 - 20k + 12k = 12,496
\][/tex]
[tex]\[
16,800 - 8k = 12,496
\][/tex]
Subtract 16,800 from both sides:
[tex]\[
-8k = 12,496 - 16,800
\][/tex]
[tex]\[
-8k = -4,304
\][/tex]
Divide by -8:
[tex]\[
k = \frac{-4,304}{-8} = 538
\][/tex]
Now, substitute [tex]\(k = 538\)[/tex] back into the first equation:
[tex]\[
a + 538 = 840
\][/tex]
[tex]\[
a = 840 - 538 = 302
\][/tex]
So, we have:
- 307 adult tickets
- 533 kids tickets
Thus, the answer is:
(E) 307 adult tickets 533 kid tickets
