Question

1.
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 observe the changes in y as x increases.

Here is the data provided:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -4 \\
7 & -7 \\
13 & -10 \\
19 & -13 \\
\hline
\end{array}
\]

First, let's calculate the differences in y as x changes:

- From \(x = 1\) to \(x = 7\):
\(y\) changes from \(-4\) to \(-7\) → change in \(y = -7 - (-4) = -3\)

- From \(x = 7\) to \(x = 13\):
\(y\) changes from \(-7\) to \(-10\) → change in \(y = -10 - (-7) = -3\)

- From \(x = 13\) to \(x = 19\):
\(y\) changes from \(-10\) to \(-13\) → change in \(y = -13 - (-10) = -3\)

Now let's examine the changes in \(x\):

- From \(x = 1\) to \(x = 7\):
change in \(x = 7 - 1 = 6\)

- From \(x = 7\) to \(x = 13\):
change in \(x = 13 - 7 = 6\)

- From \(x = 13\) to \(x = 19\):
change in \(x = 19 - 13 = 6\)

Since the change in \(y\) is consistent (\(-3\)) for a consistent change in \(x\) (\(6\)), we can conclude that the relationship is linear.

### Finding the equation of the line

We can use the slope-intercept form of the linear equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

**Step 1: Calculate the slope \(m\)**

The slope \(m\) is given by \(m = \frac{\Delta y}{\Delta x} = \frac{-3}{6} = -\frac{1}{2}\).

**Step 2: Use a point to find \(b\)**

We can use one of the points. Let's use \((1, -4)\):

\[
y = mx + b \implies -4 = -\frac{1}{2}(1) + b
\]

Solving for \(b\):

\[
-4 = -\frac{1}{2} + b \implies b = -4 + \frac{1}{2} = -\frac{8}{2} + \frac{1}{2} = -\frac{7}{2}
\]

**Final Equation:**

Thus, the equation of the line is:

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

This confirms that the relationship between \(x\) and \(y\) is indeed linear.