To determine if the relationship shown by the data is linear, we can plot the points and look for a consistent change in \(y\) values as \(x\) increases. Alternatively, we can calculate the slope using the data points.
Let's look at the given data points:
\[ \begin{align*} (1, -4) \ (7, -7) \ (13, -10) \ (19, -13) \ \end{align*} \]
We can calculate the slope \(m\) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Calculating the slope between the first two points \((1, -4)\) and \((7, -7)\):
\[ m = \frac{-7 - (-4)}{7 - 1} = \frac{-7 + 4}{6} = \frac{-3}{6} = -\frac{1}{2} \]
Next, we'll calculate the slope using the second and third points \((7, -7)\) and \((13, -10)\):
\[ m = \frac{-10 - (-7)}{13 - 7} = \frac{-10 + 7}{6} = \frac{-3}{6} = -\frac{1}{2} \]
Finally, calculating the slope between the third and fourth points \((13, -10)\) and \((19, -13)\):
\[ m = \frac{-13 - (-10)}{19 - 13} = \frac{-13 + 10}{6} = \frac{-3}{6} = -\frac{1}{2} \]
Since the slope is consistently \(-\frac{1}{2}\) for all segments, we can conclude that the relationship is linear.
Next, we can find the equation of the line using the slope-intercept form of the equation \(y = mx + b\).
We have:
- Slope \(m = -\frac{1}{2}\)
Now, we can use one of the points to solve for \(b\). Let's use the point \((1, -4)\):
\[ -4 = -\frac{1}{2}(1) + b \]
\[ -4 = -\frac{1}{2} + b \]
\[ b = -4 + \frac{1}{2} = -\frac{8}{2} + \frac{1}{2} = -\frac{7}{2} \]
The equation of the line that models this data is:
\[ y = -\frac{1}{2}x - \frac{7}{2} \]
Thus, the linear relationship can be modeled by the equation:
\[ y = -\frac{1}{2}x - \frac{7}{2} \]
1 answer