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Examine the following table that represents some points on a line.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-6 & -10 \\
\hline
-4 & -9 \\
\hline
6 & -4 \\
\hline
\end{tabular}

Which equations in point-slope form represent this line? Select all that apply.

A. [tex]$y + 4 = 2(x - 6)$[/tex]

B. [tex]$y + 4 = \frac{1}{2}(x - 6)$[/tex]

C. [tex]$y + 10 = 2(x + 6)$[/tex]

D. [tex]$y + 10 = \frac{1}{2}(x + 6)$[/tex]

E. [tex]$y - 9 = 2(x - 4)$[/tex]

F. [tex]$y + 9 = \frac{1}{2}(x + 4)$[/tex]



Answer :

GinnyAnswer
To identify which equations in point-slope form represent the line that passes through the given points, we'll follow these steps:

1. Identify Given Points:
The points given in the table are:
[tex]\[ (-6, -10), (-4, -9), (6, -4) \][/tex]

2. Calculate the Slope (m):
We use the formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points [tex]\((-6, -10)\)[/tex] and [tex]\((-4, -9)\)[/tex], we calculate:
[tex]\[ m = \frac{-9 - (-10)}{-4 - (-6)} = \frac{-9 + 10}{-4 + 6} = \frac{1}{2} \][/tex]

3. Verify Point-Slope Form Equations:
We'll use this slope [tex]\(m = \frac{1}{2}\)[/tex] and check each given point-slope form equation to see if it is satisfied by all the points.

The given equations are:
- [tex]\(y + 4 = 2(x - 6)\)[/tex]
- [tex]\(y + 4 = \frac{1}{2}(x - 6)\)[/tex]
- [tex]\(y + 10 = 2(x + 6)\)[/tex]
- [tex]\(y + 10 = \frac{1}{2}(x + 6)\)[/tex]
- [tex]\(y - 9 = 2(x - 4)\)[/tex]
- [tex]\(y + 9 = \frac{1}{2}(x + 4)\)[/tex]

We check each equation against all the given points:

For [tex]\(y + 4 = 2(x - 6)\)[/tex]:
- At [tex]\((-6, -10)\)[/tex]:
[tex]\[ -10 + 4 = 2(-6 - 6) \\ -6 \neq -24 \quad \text{(Does not satisfy)} \][/tex]

For [tex]\(y + 4 = \frac{1}{2}(x - 6)\)[/tex]:
- At [tex]\((-6, -10)\)[/tex]:
[tex]\[ -10 + 4 = \frac{1}{2}(-6 - 6) \\ -6 = -6 \quad \text{(Satisfies)} \][/tex]
- At [tex]\((-4, -9)\)[/tex]:
[tex]\[ -9 + 4 = \frac{1}{2}(-4 - 6) \\ -5 = -5 \quad \text{(Satisfies)} \][/tex]
- At [tex]\((6, -4)\)[/tex]:
[tex]\[ -4 + 4 = \frac{1}{2}(6 - 6) \\ 0 = 0 \quad \text{(Satisfies)} \][/tex]

For [tex]\(y + 10 = 2(x + 6)\)[/tex]:
- At [tex]\((-6, -10)\)[/tex]:
[tex]\[ -10 + 10 = 2(-6 + 6) \\ 0 = 0 \quad \text{(Satisfies)} \][/tex]
- At [tex]\((-4, -9)\)[/tex]:
[tex]\[ -9 + 10 = 2(-4 + 6) \\ 1 \neq 4 \quad \text{(Does not satisfy)} \][/tex]

For [tex]\(y + 10 = \frac{1}{2}(x + 6)\)[/tex]:
- At [tex]\((-6, -10)\)[/tex]:
[tex]\[ -10 + 10 = \frac{1}{2}(-6 + 6) \\ 0 = 0 \quad \text{(Satisfies)} \][/tex]
- At [tex]\((-4, -9)\)[/tex]:
[tex]\[ -9 + 10 = \frac{1}{2}(-4 + 6) \\ 1 = 1 \quad \text{(Satisfies)} \][/tex]
- At [tex]\((6, -4)\)[/tex]:
[tex]\[ -4 + 10 = \frac{1}{2}(6 + 6) \\ 6 = 6 \quad \text{(Satisfies)} \][/tex]

For [tex]\(y - 9 = 2(x - 4)\)[/tex]:
- At [tex]\((-6, -10)\)[/tex]:
[tex]\[ -10 - 9 = 2(-6 - 4) \\ -19 \neq -20 \quad \text{(Does not satisfy)} \][/tex]

For [tex]\(y + 9 = \frac{1}{2}(x + 4)\)[/tex]:
- At [tex]\((-6, -10)\)[/tex]:
[tex]\[ -10 + 9 = \frac{1}{2}(-6 + 4) \\ -1 = -1 \quad \text{(Satisfies)} \][/tex]
- At [tex]\((-4, -9)\)[/tex]:
[tex]\[ -9 + 9 = \frac{1}{2}(-4 + 4) \\ 0 = 0 \quad \text{(Satisfies)} \][/tex]
- At [tex]\((6, -4)\)[/tex]:
[tex]\[ -4 + 9 = \frac{1}{2}(6 + 4) \\ 5 = 5 \quad \text{(Satisfies)} \][/tex]

Based on the verification, the point-slope form equations that represent the line are:
[tex]\[ y + 4 = \frac{1}{2}(x - 6), \quad y + 10 = \frac{1}{2}(x + 6), \quad y + 9 = \frac{1}{2}(x + 4) \][/tex]

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