Question

neil and his friends started a country band, big hats and belt buckles, and they are about to perform their first ever concert! the concert venue will pay them a flat fee plus a fraction of the ticket sales.
this table shows the relationship between the number of concert tickets sold, x, and the amount of money (in dollars) that neil and his friends earn, y.

x (tickets)	y (dollars)
22	$95
23	$97
24	$99

according to the values in the table, do x and y have a proportional relationship?

• yes
• no

write an equation for the relationship between x and y. simplify any fractions.
$y = \square$

Explanation:

Step1: Check Proportionality

For a proportional relationship, \( \frac{y}{x} \) should be constant. Let's check the ratios:

• For \( x = 20, y = 91 \): \( \frac{91}{20}=4.55 \)
• For \( x = 22, y = 95 \): \( \frac{95}{22}\approx4.318 \)

Since the ratios are not equal, \( x \) and \( y \) are not proportional. So the answer to the first part is "no".

Step2: Find the Slope (m)

The slope \( m \) between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take \( (20,91) \) and \( (22,95) \):
\( m=\frac{95 - 91}{22 - 20}=\frac{4}{2}=2 \)

Step3: Find the y - intercept (b)

Use the point - slope form \( y - y_1=m(x - x_1) \). Using \( (20,91) \) and \( m = 2 \):
\( y-91 = 2(x - 20) \)
\( y-91=2x-40 \)
\( y=2x + 51 \)
We can verify with other points. For \( x = 23 \): \( y=2(23)+51=46 + 51=97 \) (matches the table). For \( x = 24 \): \( y=2(24)+51=48+51 = 99 \) (matches the table).

Answer:

The answer to "do \( x \) and \( y \) have a proportional relationship?" is no. The equation for the relationship is \( y = 2x+51 \)