Use the table of points to answer the following questions.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-8 & 9 \\
\hline
-4 & 6 \\
\hline
2 & [tex]$\frac{3}{2}$[/tex] \\
\hline
\end{tabular}

Part A: What is the slope from [tex]$(-8, 9)$[/tex] to [tex]$(-4, 6)$[/tex]? Show every step of your work. (1 point)

Part B: What is the slope from [tex]$(-4, 6)$[/tex] to [tex]$\left(2, \frac{3}{2}\right)$[/tex]? Show every step of your work. (1 point)

Part C: What do the slopes from Parts A and B tell you about the relationship between all the points in the table? (2 points)



Answer :

Let's solve the problem step-by-step.

### Part A: Calculating the Slope from [tex]\((-8, 9)\)[/tex] to [tex]\((-4, 6)\)[/tex]

To find the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we use the slope formula:

[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

For points [tex]\((-8, 9)\)[/tex] and [tex]\((-4, 6)\)[/tex]:
[tex]\[ x_1 = -8, \quad y_1 = 9, \quad x_2 = -4, \quad y_2 = 6 \][/tex]

Plug these values into the formula:
[tex]\[ \text{slope}_a = \frac{6 - 9}{-4 - (-8)} = \frac{6 - 9}{-4 + 8} = \frac{-3}{4} = -0.75 \][/tex]

So, the slope from [tex]\((-8, 9)\)[/tex] to [tex]\((-4, 6)\)[/tex] is [tex]\(\mathbf{-0.75}\)[/tex].

### Part B: Calculating the Slope from [tex]\((-4, 6)\)[/tex] to [tex]\(\left(2, \frac{3}{2}\right)\)[/tex]

Using the same slope formula for points [tex]\((-4, 6)\)[/tex] and [tex]\(\left(2, \frac{3}{2}\right)\)[/tex]:
[tex]\[ x_1 = -4, \quad y_1 = 6, \quad x_2 = 2, \quad y_2 = \frac{3}{2} \][/tex]

Plug these values into the formula:
[tex]\[ \text{slope}_b = \frac{\frac{3}{2} - 6}{2 - (-4)} = \frac{\frac{3}{2} - 6}{2 + 4} = \frac{\frac{3}{2} - \frac{12}{2}}{6} = \frac{\frac{3 - 12}{2}}{6} = \frac{\frac{-9}{2}}{6} = \frac{-9}{12} = \frac{-3}{4} = -0.75 \][/tex]

So, the slope from [tex]\((-4, 6)\)[/tex] to [tex]\(\left(2, \frac{3}{2}\right)\)[/tex] is [tex]\(\mathbf{-0.75}\)[/tex].

### Part C: Relationship Between the Points

We have calculated the slopes from the previous segments:
- Slope from [tex]\((-8, 9)\)[/tex] to [tex]\((-4, 6)\)[/tex] is [tex]\(-0.75\)[/tex]
- Slope from [tex]\((-4, 6)\)[/tex] to [tex]\(\left(2, \frac{3}{2}\right)\)[/tex] is [tex]\(-0.75\)[/tex]

Since both slopes are equal ([tex]\(-0.75\)[/tex]), all three points lie on the same straight line. This indicates that the points [tex]\((-8, 9)\)[/tex], [tex]\((-4, 6)\)[/tex], and [tex]\(\left(2, \frac{3}{2}\right)\)[/tex] are collinear.

Therefore, the relationship between the points is that they all lie on the same straight line.