Question

The equation 𝑦 = 0.25 𝑥 y=0.25x describes a proportional relationship between 𝑥 x and 𝑦 y. What is the constant of proportionality?
k=___

Answer 1

Does this table show a proportional relationship? If yes, what is the constant of proportionality?
x 3 6 9 12
y 12 24 45 60

No, the quantities are not proportional.
Yes, 4 is the constant of proportionality.
Yes, 1/4 is the constant of proportionality.
Yes, 5 is the constant of proportionality.

Answer 2

what is the ratio if a slope is going from (-1, -3) to (4, -2) in fraction form?

Answer 3

The equation \( y = 0.25x \) shows that \( y \) is directly proportional to \( x \). In a proportional relationship of the form \( y = kx \), the constant of proportionality \( k \) is the coefficient of \( x \).

In this case, the constant of proportionality is:

\[
k = 0.25
\]

So, \( k = 0.25 \).

Answer 4

To determine if the relationship between \( x \) and \( y \) is proportional, we need to check if the ratio \( \frac{y}{x} \) remains constant for all pairs of \( (x, y) \).

Let's calculate the ratio \( \frac{y}{x} \) for each pair:

1. For \( x = 3, y = 12 \):

\[
\frac{y}{x} = \frac{12}{3} = 4
\]

2. For \( x = 6, y = 24 \):

\[
\frac{y}{x} = \frac{24}{6} = 4
\]

3. For \( x = 9, y = 45 \):

\[
\frac{y}{x} = \frac{45}{9} = 5
\]

4. For \( x = 12, y = 60 \):

\[
\frac{y}{x} = \frac{60}{12} = 5
\]

The ratios are:
- For \( (3, 12) \): \( 4 \)
- For \( (6, 24) \): \( 4 \)
- For \( (9, 45) \): \( 5 \)
- For \( (12, 60) \): \( 5 \)

Since the ratios are not constant (4 for the first two pairs and 5 for the last two), the relationship is not proportional.

Therefore, the answer is:

**No, the quantities are not proportional.**

Answer 5

To find the slope of a line between two points \((-1, -3)\) and \( (4, -2) \), we can use the slope formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here:
- \( (x_1, y_1) = (-1, -3) \)
- \( (x_2, y_2) = (4, -2) \)

Now, substituting the values into the slope formula:

\[
m = \frac{-2 - (-3)}{4 - (-1)} = \frac{-2 + 3}{4 + 1} = \frac{1}{5}
\]

So, the slope of the line, in fraction form, is:

\[
\frac{1}{5}
\]