Answer :
To determine which ordered pairs could be points on a line parallel to the line that contains the points [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex], we need to find the slope of the line formed by these two points, and then check for other pairs of points that have the same slope.
### Step 1: Find the Slope of the Line through [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex]
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in [tex]\((x_1, y_1) = (3, 4)\)[/tex] and [tex]\((x_2, y_2) = (-2, 2)\)[/tex]:
[tex]\[ m = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = 0.4 \][/tex]
So, the slope of the line through the points [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] is [tex]\(0.4\)[/tex].
### Step 2: Check Each Pair to See if They Have the Same Slope
### Pair 1: [tex]\((-2, -5)\)[/tex] and [tex]\((-7, -3)\)[/tex]
[tex]\[ m = \frac{-3 - (-5)}{-7 - (-2)} = \frac{-3 + 5}{-7 + 2} = \frac{2}{-5} = -0.4 \][/tex]
This pair does not have the same slope.
### Pair 2: [tex]\((-1, 1)\)[/tex] and [tex]\((-6, -1)\)[/tex]
[tex]\[ m = \frac{-1 - 1}{-6 - (-1)} = \frac{-1 - 1}{-6 + 1} = \frac{-2}{-5} = 0.4 \][/tex]
This pair has the same slope.
### Pair 3: [tex]\((0, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
[tex]\[ m = \frac{5 - 0}{2 - 0} = \frac{5}{2} = 2.5 \][/tex]
This pair does not have the same slope.
### Pair 4: [tex]\((1, 0)\)[/tex] and [tex]\((6, 2)\)[/tex]
[tex]\[ m = \frac{2 - 0}{6 - 1} = \frac{2}{5} = 0.4 \][/tex]
This pair has the same slope.
### Pair 5: [tex]\((3, 0)\)[/tex] and [tex]\((8, 2)\)[/tex]
[tex]\[ m = \frac{2 - 0}{8 - 3} = \frac{2}{5} = 0.4 \][/tex]
This pair has the same slope.
### Conclusion
The pairs that could be points on a line parallel to the line containing the points [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] are:
[tex]\[ (-1, 1) \text{ and } (-6, -1) \][/tex]
[tex]\[ (1, 0) \text{ and } (6, 2) \][/tex]
[tex]\[ (3, 0) \text{ and } (8, 2) \][/tex]
So, these are the ordered pairs that have lines with the same slope of [tex]\(0.4\)[/tex].
### Step 1: Find the Slope of the Line through [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex]
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in [tex]\((x_1, y_1) = (3, 4)\)[/tex] and [tex]\((x_2, y_2) = (-2, 2)\)[/tex]:
[tex]\[ m = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = 0.4 \][/tex]
So, the slope of the line through the points [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] is [tex]\(0.4\)[/tex].
### Step 2: Check Each Pair to See if They Have the Same Slope
### Pair 1: [tex]\((-2, -5)\)[/tex] and [tex]\((-7, -3)\)[/tex]
[tex]\[ m = \frac{-3 - (-5)}{-7 - (-2)} = \frac{-3 + 5}{-7 + 2} = \frac{2}{-5} = -0.4 \][/tex]
This pair does not have the same slope.
### Pair 2: [tex]\((-1, 1)\)[/tex] and [tex]\((-6, -1)\)[/tex]
[tex]\[ m = \frac{-1 - 1}{-6 - (-1)} = \frac{-1 - 1}{-6 + 1} = \frac{-2}{-5} = 0.4 \][/tex]
This pair has the same slope.
### Pair 3: [tex]\((0, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
[tex]\[ m = \frac{5 - 0}{2 - 0} = \frac{5}{2} = 2.5 \][/tex]
This pair does not have the same slope.
### Pair 4: [tex]\((1, 0)\)[/tex] and [tex]\((6, 2)\)[/tex]
[tex]\[ m = \frac{2 - 0}{6 - 1} = \frac{2}{5} = 0.4 \][/tex]
This pair has the same slope.
### Pair 5: [tex]\((3, 0)\)[/tex] and [tex]\((8, 2)\)[/tex]
[tex]\[ m = \frac{2 - 0}{8 - 3} = \frac{2}{5} = 0.4 \][/tex]
This pair has the same slope.
### Conclusion
The pairs that could be points on a line parallel to the line containing the points [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] are:
[tex]\[ (-1, 1) \text{ and } (-6, -1) \][/tex]
[tex]\[ (1, 0) \text{ and } (6, 2) \][/tex]
[tex]\[ (3, 0) \text{ and } (8, 2) \][/tex]
So, these are the ordered pairs that have lines with the same slope of [tex]\(0.4\)[/tex].
