Let's determine if line and line k are parallel by calculating their slopes. The slope of a line is a measure of how steep the line is, and it is defined as the vertical change (the change in the y-coordinates) divided by the horizontal change (the change in the x-coordinates) between two points on the line.
For line , we will use the points A(-2, 2) and B(-1, 0).
The slope (m) of line can be calculated using the formula:
[tex]\[ m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the coordinates of A and B, we get:
[tex]\[ m = \frac{0 - 2}{-1 - (-2)} = \frac{-2}{1} = -2 \][/tex]
Now, let's calculate the slope of line k using the points C(0, 4) and D(1, 2).
The slope (m_k) of line k is calculated in the same manner:
[tex]\[ m_k = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the coordinates of C and D, we get:
[tex]\[ m_k = \frac{2 - 4}{1 - 0} = \frac{-2}{1} = -2 \][/tex]
Now we have calculated the slopes of both lines:
- The slope of line is -2.
- The slope of line k is -2 as well.
Two lines are parallel if and only if their slopes are equal and they are not coincident (meaning not the exact same line). Since the slopes of line and line k are both -2, we can conclude that lines and k are parallel.
