Question

math - 8th grade
34 a teacher buys adult and child tickets for a class field trip.

• each adult ticket costs $12, and each child ticket costs $8.
• the teacher spends a total of $148 and buys 17 total tickets.

how many child tickets does the teacher buy?
a 3 child tickets
b 5 child tickets
c 12 child tickets
d 14 child tickets

1. for this item, enter the answer in the space provided.

consider the linear function represented by the table.

x	y
0	-3
2	-2
4	-1
6	0

based on the table, what is the rate of change of the linear function?

Explanation:

Question 34

Step1: Define variables

Let \( x \) be the number of adult tickets and \( y \) be the number of child tickets. We know two equations:

1. \( x + y = 17 \) (total tickets)
2. \( 12x + 8y = 148 \) (total cost)

From the first equation, we can express \( x = 17 - y \).

Step2: Substitute and solve

Substitute \( x = 17 - y \) into the second equation:
\( 12(17 - y) + 8y = 148 \)
Expand: \( 204 - 12y + 8y = 148 \)
Simplify: \( 204 - 4y = 148 \)
Subtract 204: \( -4y = 148 - 204 = -56 \)
Divide by -4: \( y = \frac{-56}{-4} = 14 \)? Wait, no, wait. Wait, let's check again. Wait, maybe I made a mistake. Wait, let's re - calculate:

Wait, \( 12(17 - y)+8y = 148 \)

\( 204-12y + 8y=148 \)

\( 204 - 4y=148 \)

\( - 4y=148 - 204=-56 \)

\( y = 14 \)? But wait, let's check the options. Option D is 14 child tickets. But let's verify with the total number of tickets. If \( y = 14 \), then \( x=17 - 14 = 3 \). Then total cost: \( 12\times3+8\times14=36 + 112 = 148 \). Yes, that works. Wait, but let's check the options again. The options are A. 3, B.5, C.12, D.14. So the answer is D.

Wait, but maybe I misread the option. Wait, the question is "How many child tickets does the teacher buy?". So according to the calculation, \( y = 14 \), which is option D.

Question 35

Step1: Recall rate of change formula

The rate of change (slope) of a linear function is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \)

Step2: Choose two points

Let's take two points from the table, say \( (0,-3) \) and \( (2,-2) \)

Step3: Calculate slope

Using the formula \( m=\frac{-2-(-3)}{2 - 0}=\frac{-2 + 3}{2}=\frac{1}{2}=0.5 \)

We can also check with other points, like \( (2,-2) \) and \( (4,-1) \): \( m=\frac{-1-(-2)}{4 - 2}=\frac{-1 + 2}{2}=\frac{1}{2} \)

Or \( (4,-1) \) and \( (6,0) \): \( m=\frac{0-(-1)}{6 - 4}=\frac{1}{2} \)

So the rate of change is \( \frac{1}{2} \) or 0.5.

Question 34 Answer: D. 14 child tickets

Question 35 Answer: \(\frac{1}{2}\) (or 0.5)

Answer:

Step1: Recall rate of change formula

The rate of change (slope) of a linear function is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \)

Step2: Choose two points

Let's take two points from the table, say \( (0,-3) \) and \( (2,-2) \)

Step3: Calculate slope

Using the formula \( m=\frac{-2-(-3)}{2 - 0}=\frac{-2 + 3}{2}=\frac{1}{2}=0.5 \)

We can also check with other points, like \( (2,-2) \) and \( (4,-1) \): \( m=\frac{-1-(-2)}{4 - 2}=\frac{-1 + 2}{2}=\frac{1}{2} \)

Or \( (4,-1) \) and \( (6,0) \): \( m=\frac{0-(-1)}{6 - 4}=\frac{1}{2} \)

So the rate of change is \( \frac{1}{2} \) or 0.5.

Question 34 Answer: D. 14 child tickets

Question 35 Answer: \(\frac{1}{2}\) (or 0.5)