To determine if the lines given by the equations [tex]\( 2x + 9y = -7 \)[/tex] and [tex]\( 9x - 2y = -2 \)[/tex] are parallel, perpendicular, or neither, we first need to find the slopes of these lines. Here's how we can do it step-by-step:
### Step 1: Rewrite the Equations in Slope-Intercept Form
The slope-intercept form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
For the first equation [tex]\( 2x + 9y = -7 \)[/tex]:
1. Isolate [tex]\( y \)[/tex] on one side of the equation:
[tex]\[
9y = -2x - 7
\][/tex]
2. Divide all terms by 9 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{2}{9}x - \frac{7}{9}
\][/tex]
The slope ([tex]\( m_1 \)[/tex]) of the first line is [tex]\( -\frac{2}{9} \)[/tex].
For the second equation [tex]\( 9x - 2y = -2 \)[/tex]:
1. Isolate [tex]\( y \)[/tex] on one side of the equation:
[tex]\[
-2y = -9x - 2
\][/tex]
2. Divide all terms by -2 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{9}{2}x + 1
\][/tex]
The slope ([tex]\( m_2 \)[/tex]) of the second line is [tex]\( \frac{9}{2} \)[/tex].
### Step 2: Compare the Slopes
1. The slope of the first line is [tex]\( -\frac{2}{9} \)[/tex].
2. The slope of the second line is [tex]\( \frac{9}{2} \)[/tex].
### Step 3: Determine the Relationship Between the Lines
- Parallel Lines: Two lines are parallel if their slopes are equal, [tex]\( m_1 = m_2 \)[/tex].
[tex]\[
-\frac{2}{9} \neq \frac{9}{2}
\][/tex]
Therefore, the lines are not parallel.
- Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\( -1 \)[/tex], [tex]\( m_1 \cdot m_2 = -1 \)[/tex].
[tex]\[
-\frac{2}{9} \cdot \frac{9}{2} = -1
\][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], the lines are perpendicular.
- Neither: If the lines are neither parallel nor perpendicular, they would not satisfy the above conditions.
### Conclusion
Since the product of the slopes [tex]\(-\frac{2}{9}\)[/tex] and [tex]\(\frac{9}{2}\)[/tex] is [tex]\(-1\)[/tex], the lines [tex]\( 2x + 9y = -7 \)[/tex] and [tex]\( 9x - 2y = -2 \)[/tex] are perpendicular.
