Answer :
To determine the type of slope for the line that passes through each pair of given points, we can follow a step-by-step process. Recall that the slope of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let's analyze each pair of points:
1. Points [tex]\((-7, 8)\)[/tex] and [tex]\((-7, 0)\)[/tex]
- Here, [tex]\(x_1 = x_2 = -7\)[/tex]. When the x-coordinates are the same, the line is vertical.
- The slope of a vertical line is undefined.
- Therefore, the slope is undefined.
2. Points [tex]\((3, 5)\)[/tex] and [tex]\((-1, 2)\)[/tex]
- Calculate the slope: [tex]\(\frac{2 - 5}{-1 - 3} = \frac{-3}{-4} = \frac{3}{4}\)[/tex].
- Because the result is positive, the slope is positive.
- Therefore, the slope is positive.
3. Points [tex]\((6, -3)\)[/tex] and [tex]\((-4, -3)\)[/tex]
- Here, [tex]\(y_1 = y_2 = -3\)[/tex]. When the y-coordinates are the same, the line is horizontal.
- The slope of a horizontal line is zero.
- Therefore, the slope is zero.
4. Points [tex]\((2, 4)\)[/tex] and [tex]\((5, 1)\)[/tex]
- Calculate the slope: [tex]\(\frac{1 - 4}{5 - 2} = \frac{-3}{3} = -1\)[/tex].
- Because the result is negative, the slope is negative.
- Therefore, the slope is negative.
To summarize:
- The slope for the line through points [tex]\((-7, 8)\)[/tex] and [tex]\((-7, 0)\)[/tex] is undefined.
- The slope for the line through points [tex]\((3, 5)\)[/tex] and [tex]\((-1, 2)\)[/tex] is positive.
- The slope for the line through points [tex]\((6, -3)\)[/tex] and [tex]\((-4, -3)\)[/tex] is zero.
- The slope for the line through points [tex]\((2, 4)\)[/tex] and [tex]\((5, 1)\)[/tex] is negative.
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let's analyze each pair of points:
1. Points [tex]\((-7, 8)\)[/tex] and [tex]\((-7, 0)\)[/tex]
- Here, [tex]\(x_1 = x_2 = -7\)[/tex]. When the x-coordinates are the same, the line is vertical.
- The slope of a vertical line is undefined.
- Therefore, the slope is undefined.
2. Points [tex]\((3, 5)\)[/tex] and [tex]\((-1, 2)\)[/tex]
- Calculate the slope: [tex]\(\frac{2 - 5}{-1 - 3} = \frac{-3}{-4} = \frac{3}{4}\)[/tex].
- Because the result is positive, the slope is positive.
- Therefore, the slope is positive.
3. Points [tex]\((6, -3)\)[/tex] and [tex]\((-4, -3)\)[/tex]
- Here, [tex]\(y_1 = y_2 = -3\)[/tex]. When the y-coordinates are the same, the line is horizontal.
- The slope of a horizontal line is zero.
- Therefore, the slope is zero.
4. Points [tex]\((2, 4)\)[/tex] and [tex]\((5, 1)\)[/tex]
- Calculate the slope: [tex]\(\frac{1 - 4}{5 - 2} = \frac{-3}{3} = -1\)[/tex].
- Because the result is negative, the slope is negative.
- Therefore, the slope is negative.
To summarize:
- The slope for the line through points [tex]\((-7, 8)\)[/tex] and [tex]\((-7, 0)\)[/tex] is undefined.
- The slope for the line through points [tex]\((3, 5)\)[/tex] and [tex]\((-1, 2)\)[/tex] is positive.
- The slope for the line through points [tex]\((6, -3)\)[/tex] and [tex]\((-4, -3)\)[/tex] is zero.
- The slope for the line through points [tex]\((2, 4)\)[/tex] and [tex]\((5, 1)\)[/tex] is negative.
