Question

Theequationoflinemis
y
=
5
9
x
+
5
3
. The
equationoflinenis
y
=
–9
5
x
+ 8
. Are
linemandlinenparallelorperpendicular?

Answer 1

To determine whether the lines are parallel or perpendicular, we need to compare their slopes.

The equations of the lines are:

1. Line m: \( y = \frac{5}{9}x + \frac{5}{3} \)
2. Line n: \( y = -\frac{9}{5}x + 8 \)

### Step 1: Identify the slopes

- The slope of line m (\( m_1 \)) is \( \frac{5}{9} \).
- The slope of line n (\( m_2 \)) is \( -\frac{9}{5} \).

### Step 2: Check for parallelism

Two lines are parallel if their slopes are equal. Here:

\[
m_1 = \frac{5}{9} \quad \text{and} \quad m_2 = -\frac{9}{5}
\]

Since \( \frac{5}{9} \neq -\frac{9}{5} \), the lines are **not parallel**.

### Step 3: Check for perpendicularity

Two lines are perpendicular if the product of their slopes is -1:

\[
m_1 \times m_2 = \frac{5}{9} \times -\frac{9}{5}
\]

Calculating the product:

\[
\frac{5}{9} \times -\frac{9}{5} = -1
\]

Since the product of the slopes is -1, the lines are **perpendicular**.

### Conclusion

The lines are perpendicular.