Question

Theequationoflinemis
y
=
23
x
−
2
85
. The
equationoflinenisy = 23x + 60
. Are
linemandlinenparallelorperpendicular?

Answer 1

To determine whether the lines represented by the equations \( y = \frac{23}{2} x - \frac{285}{100} \) and \( y = 23x + 60 \) are parallel or perpendicular, we first need to identify their slopes.

1. **Finding the slope of the first line:**
The equation given is \( y = \frac{23}{2} x - \frac{285}{100} \).
- The slope (m₁) of this line is \( \frac{23}{2} \).

2. **Finding the slope of the second line:**
The equation is \( y = 23x + 60 \).
- The slope (m₂) of this line is \( 23 \).

3. **Comparing the slopes:**
Because the two slopes are compared, we check:
- For the lines to be **parallel**, their slopes must be equal, which means \( m₁ = m₂ \).
- For the lines to be **perpendicular**, the product of their slopes must equal \(-1\), which means \( m₁ \cdot m₂ = -1 \).

Let's compute the product of the slopes:
\[
\frac{23}{2} \cdot 23 = \frac{23 \cdot 23}{2} = \frac{529}{2}
\]

Since \( \frac{529}{2} \) is not equal to \(-1\), the lines are not perpendicular.

Next, let's compare the slopes:
\[
\frac{23}{2} \text{ (which is 11.5) is not equal to } 23.
\]

Since \( \frac{23}{2} \) is not equal to \( 23 \), the lines are not parallel.

**Conclusion:**
The lines are neither parallel nor perpendicular.