Question

Suppose you were given this table. How would you write the equation of the graph of this relationship?
Which statements are true? Select all that apply.



x y
6 3
12 6
18 9
24 12


A.


The ratio of x to y is the slope.

B.


The ratio of y to x is the slope.

C.


The equation of the graph is y = 2x.

D.


The equation of the graph is y = 12
x.

Answer 1

To analyze the table provided, let's first determine the relationship between \( x \) and \( y \) based on the values given:

\[
\begin{align*}
x & : 6, 12, 18, 24 \\
y & : 3, 6, 9, 12
\end{align*}
\]

Now let's determine if there is a constant ratio or a linear relationship between \( x \) and \( y \).

To find the slope, we take several pairs of points to determine the change in \( y \) for a corresponding change in \( x \):

1. Using the points (6, 3) and (12, 6):
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 3}{12 - 6} = \frac{3}{6} = \frac{1}{2}
\]

2. Using the points (12, 6) and (18, 9):
\[
\text{slope} = \frac{9 - 6}{18 - 12} = \frac{3}{6} = \frac{1}{2}
\]

3. Using the points (18, 9) and (24, 12):
\[
\text{slope} = \frac{12 - 9}{24 - 18} = \frac{3}{6} = \frac{1}{2}
\]

Each calculation shows that the slope is consistently \( \frac{1}{2} \).

Next, we can express the linear equation of the relationship. Since the slope (\( m \)) is \( \frac{1}{2} \) and it looks like \( y \) is half of \( x \), we can express this relationship as:
\[
y = \frac{1}{2}x
\]

However, if rewriting this equation in a more common form, we can rearrange:
\[
y = \frac{1}{2}x \implies 2y = x
\]
So, the equation could also be expressed differently but does not reflect the options given in your statements.

### Evaluating the Given Statements

A. **The ratio of x to y is the slope.**
This is **false**. The slope is calculated as the change in \( y \) over the change in \( x \), not the ratio of \( x \) to \( y \).

B. **The ratio of y to x is the slope.**
This is **true**. If you take any \( y \) value and divide by its corresponding \( x \) value, it reflects the slope as \( y/x = \frac{1}{2} \).

C. **The equation of the graph is y = 2x.**
This is **false**. The correct equation is \( y = \frac{1}{2}x \).

D. **The equation of the graph is y = 12x.**
This is **false**. The correct equation is \( y = \frac{1}{2}x \).

### Summary of True Statements
- B is the only true statement.

Thus, the statements that are true are:

- **B**: The ratio of y to x is the slope.