Question: Given below are descriptions of two lines.
Line 1: Goes through $(5,15)$ and $(-1,3)$
Line 2: Goes through $(13,-81)$ and $(14,-88)$
The slope of Line 1 is $m =$
The slope of Line 2 is $m =$
Finally, which of the following is true?
Line 1 is parallel to Line 2.
Line 1 is perpendicular to Line 2
Line 1 is neither parallel nor perpendicular to Line 2
1. **State the problem:** We need to find the slopes of two lines given two points on each line and then determine the relationship between the lines (parallel, perpendicular, or neither).
2. **Formula for slope:** The slope $m$ of a line passing through points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
3. **Calculate slope of Line 1:** Points are $(5,15)$ and $(-1,3)$.
$$m_1 = \frac{3 - 15}{-1 - 5} = \frac{-12}{-6}$$
Cancel common factors:
$$m_1 = \frac{\cancel{-12}}{\cancel{-6}} = 2$$
So, slope of Line 1 is $m_1 = 2$.
4. **Calculate slope of Line 2:** Points are $(13,-81)$ and $(14,-88)$.
$$m_2 = \frac{-88 - (-81)}{14 - 13} = \frac{-88 + 81}{1} = \frac{-7}{1}$$
No cancellation needed.
So, slope of Line 2 is $m_2 = -7$.
5. **Determine relationship:**
- Lines are parallel if their slopes are equal: $m_1 = m_2$.
- Lines are perpendicular if the product of their slopes is $-1$: $m_1 \times m_2 = -1$.
Calculate product:
$$2 \times (-7) = -14 \neq -1$$
Since $m_1 \neq m_2$ and $m_1 \times m_2 \neq -1$, the lines are neither parallel nor perpendicular.
**Final answer:**
- Slope of Line 1 is $2$.
- Slope of Line 2 is $-7$.
- Line 1 is neither parallel nor perpendicular to Line 2.