Question

Which ordered pairs could be points on a line parallel to the line that contains (3, 4) and (–2, 2)? Check all that apply.

(–2, –5) and (–7, –3)
(–1, 1) and (–6, –1)
(0, 0) and (2, 5)
(1, 0) and (6, 2)
(3, 0) and (8, 2)

Answer 1

To determine the ordered pairs that could be points on a line parallel to the line that contains the points (3, 4) and (–2, 2), we first need to find the slope of the line passing through those two points.

The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Assigning the points as follows:
- \((x_1, y_1) = (3, 4)\)
- \((x_2, y_2) = (-2, 2)\)

We calculate the slope:

\[
m = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = \frac{2}{5}
\]

A line parallel to this line will have the same slope of \(\frac{2}{5}\).

Next, to determine which of the given pairs form a line with this slope, we calculate the slope for each pair of points. A line defined by two points \((x_1, y_1)\) and \((x_2, y_2)\) will be parallel if its slope is also \(\frac{2}{5}\).

Now, we will check each option:

1. **(–2, –5) and (–7, –3)**:
\[
m = \frac{-3 - (-5)}{-7 - (-2)} = \frac{2}{-5} = -\frac{2}{5} \quad (not\ a\ match)
\]

2. **(–1, 1) and (–6, –1)**:
\[
m = \frac{-1 - 1}{-6 - (-1)} = \frac{-2}{-5} = \frac{2}{5} \quad (match)
\]

3. **(0, 0) and (2, 5)**:
\[
m = \frac{5 - 0}{2 - 0} = \frac{5}{2} \quad (not\ a\ match)
\]

4. **(1, 0) and (6, 2)**:
\[
m = \frac{2 - 0}{6 - 1} = \frac{2}{5} \quad (match)
\]

5. **(3, 0) and (8, 2)**:
\[
m = \frac{2 - 0}{8 - 3} = \frac{2}{5} \quad (match)
\]

The ordered pairs that could be points on a line parallel to the line containing (3, 4) and (–2, 2) are:
- (–1, 1) and (–6, –1)
- (1, 0) and (6, 2)
- (3, 0) and (8, 2)

Thus, the valid options are:
- (–1, 1) and (–6, –1)
- (1, 0) and (6, 2)
- (3, 0) and (8, 2)