Question

13.
Is the relationship shown by the data linear? If so, model the data with an equation.

x y
1 –4
7 –7
13 –10
19 –13

Answer 1

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:

\[
y = -\frac{1}{2}x - \frac{7}{2}
\]