Question
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
The relationship is linear; y – 1 = –2(x + 4).
The relationship is linear; y + 4 = (x – 1).
The relationship is linear; y + 4 = 2(x – 1).
The relationship is not linear.
x y
1 –4
7 –7
13 –10
19 –13
The relationship is linear; y – 1 = –2(x + 4).
The relationship is linear; y + 4 = (x – 1).
The relationship is linear; y + 4 = 2(x – 1).
The relationship is not linear.
Answers
GPT-4o mini
To determine if the relationship given by the data is linear, we can analyze the changes in the values of \(y\) as \(x\) changes.
The data points are:
- \( (1, -4) \)
- \( (7, -7) \)
- \( (13, -10) \)
- \( (19, -13) \)
Now, let's look for the change in \(y\) as \(x\) increases:
1. From \(x = 1\) to \(x = 7\):
- \(y\) changes from \(-4\) to \(-7\) → change in \(y = -7 - (-4) = -3\)
- change in \(x = 7 - 1 = 6\)
2. From \(x = 7\) to \(x = 13\):
- \(y\) changes from \(-7\) to \(-10\) → change in \(y = -10 - (-7) = -3\)
- change in \(x = 13 - 7 = 6\)
3. From \(x = 13\) to \(x = 19\):
- \(y\) changes from \(-10\) to \(-13\) → change in \(y = -13 - (-10) = -3\)
- change in \(x = 19 - 13 = 6\)
Notice that for each increase of \(6\) in \(x\), \(y\) decreases by \(3\). This consistent change indicates that the relationship is linear.
Now, we can find the slope \(m\) of the linear relationship:
\[
m = \frac{\Delta y}{\Delta x} = \frac{-3}{6} = -\frac{1}{2}
\]
Now we apply the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\). Using one of the points, say \((1, -4)\):
\[
y - (-4) = -\frac{1}{2}(x - 1)
\]
Simplifying this gives:
\[
y + 4 = -\frac{1}{2}(x - 1)
\]
We can rearrange it into slope-intercept form \(y = mx + b\).
To get the equation into the form you presented in the choices, let's further simplify:
Expanding the right side:
\[
y + 4 = -\frac{1}{2}x + \frac{1}{2}
\]
Subtracting 4 from both sides:
\[
y = -\frac{1}{2}x + \frac{1}{2} - 4
\]
\[
y = -\frac{1}{2}x - \frac{7}{2}
\]
This linearly represents the relationship.
So the relationship is linear, but the equations presented in your options seem to not match my derived equation. However, since the computations show a linear relationship, and I have confirmed a consistent change, the summary is correct:
**The relationship is linear; the slope \(m\) = -1/2.**
If the options were to be consistent with that slope, we would need to choose an option that reflects a linear equation having a slope of -1/2 and potentially a \(y\)-intercept adjustment, which I have derived above. Please check the choices you have to find the one that aligns closely with this concept. Otherwise, you'll likely need to derive based on the relationships presented.
The data points are:
- \( (1, -4) \)
- \( (7, -7) \)
- \( (13, -10) \)
- \( (19, -13) \)
Now, let's look for the change in \(y\) as \(x\) increases:
1. From \(x = 1\) to \(x = 7\):
- \(y\) changes from \(-4\) to \(-7\) → change in \(y = -7 - (-4) = -3\)
- change in \(x = 7 - 1 = 6\)
2. From \(x = 7\) to \(x = 13\):
- \(y\) changes from \(-7\) to \(-10\) → change in \(y = -10 - (-7) = -3\)
- change in \(x = 13 - 7 = 6\)
3. From \(x = 13\) to \(x = 19\):
- \(y\) changes from \(-10\) to \(-13\) → change in \(y = -13 - (-10) = -3\)
- change in \(x = 19 - 13 = 6\)
Notice that for each increase of \(6\) in \(x\), \(y\) decreases by \(3\). This consistent change indicates that the relationship is linear.
Now, we can find the slope \(m\) of the linear relationship:
\[
m = \frac{\Delta y}{\Delta x} = \frac{-3}{6} = -\frac{1}{2}
\]
Now we apply the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\). Using one of the points, say \((1, -4)\):
\[
y - (-4) = -\frac{1}{2}(x - 1)
\]
Simplifying this gives:
\[
y + 4 = -\frac{1}{2}(x - 1)
\]
We can rearrange it into slope-intercept form \(y = mx + b\).
To get the equation into the form you presented in the choices, let's further simplify:
Expanding the right side:
\[
y + 4 = -\frac{1}{2}x + \frac{1}{2}
\]
Subtracting 4 from both sides:
\[
y = -\frac{1}{2}x + \frac{1}{2} - 4
\]
\[
y = -\frac{1}{2}x - \frac{7}{2}
\]
This linearly represents the relationship.
So the relationship is linear, but the equations presented in your options seem to not match my derived equation. However, since the computations show a linear relationship, and I have confirmed a consistent change, the summary is correct:
**The relationship is linear; the slope \(m\) = -1/2.**
If the options were to be consistent with that slope, we would need to choose an option that reflects a linear equation having a slope of -1/2 and potentially a \(y\)-intercept adjustment, which I have derived above. Please check the choices you have to find the one that aligns closely with this concept. Otherwise, you'll likely need to derive based on the relationships presented.