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 \( x \):
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Using the points (6, 3) and (12, 6): \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 3}{12 - 6} = \frac{3}{6} = \frac{1}{2} \]
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Using the points (12, 6) and (18, 9): \[ \text{slope} = \frac{9 - 6}{18 - 12} = \frac{3}{6} = \frac{1}{2} \]
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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.