Let's solve the problem step by step:
### Step 1: Calculate the differences between the points
Given the data points [tex]\((-6, -6)\)[/tex], [tex]\((-2, -4)\)[/tex], and [tex]\( (4, -1) \)[/tex]:
- Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] between the first and second points:
- [tex]\(\Delta x_1 = x_2 - x_1 = -2 - (-6) = 4\)[/tex]
- [tex]\(\Delta y_1 = y_2 - y_1 = -4 - (-6) = 2\)[/tex]
- Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] between the second and third points:
- [tex]\(\Delta x_2 = x_3 - x_2 = 4 - (-2) = 6\)[/tex]
- [tex]\(\Delta y_2 = y_3 - y_2 = -1 - (-4) = 3\)[/tex]
### Step 2: Calculate the slopes between the points
- Find the slope between the first and second points using the formula [tex]\( \text{slope} = \frac{\Delta y}{\Delta x} \)[/tex]:
- [tex]\( \text{slope}_1 = \frac{\Delta y_1}{\Delta x_1} = \frac{2}{4} = 0.5 \)[/tex]
- Find the slope between the second and third points using the same formula:
- [tex]\( \text{slope}_2 = \frac{\Delta y_2}{\Delta x_2} = \frac{3}{6} = 0.5 \)[/tex]
### Conclusion
The slopes between the given points are constant, and they both equal [tex]\(0.5\)[/tex].
Thus, the slope of the line passing through the given points [tex]\((-6, -6)\)[/tex], [tex]\((-2, -4)\)[/tex], and [tex]\( (4, -1)\)[/tex] is [tex]\(0.5\)[/tex]. This finding indicates that the points lie on a straight line with the slope of [tex]\(0.5\)[/tex].
