To determine which line is perpendicular to [tex]\( y = \frac{4}{9}x - 7 \)[/tex], we need to follow the steps below:
1. Determine the slope of the given line.
2. Find the slope of the line that would be perpendicular to it.
3. Identify which of the given lines has this slope.
### Step 1: Determine the slope of the given line.
The given line equation is:
[tex]\[ y = \frac{4}{9}x - 7 \][/tex]
The slope ([tex]\( m \)[/tex]) of the line [tex]\( y = mx + b \)[/tex] is the coefficient of [tex]\( x \)[/tex]. Therefore, the slope [tex]\( m \)[/tex] of the given line is:
[tex]\[ m = \frac{4}{9} \][/tex]
### Step 2: Find the slope of the perpendicular line.
The slope of any line perpendicular to another is the negative reciprocal of the slope of the original line. The negative reciprocal of [tex]\( \frac{4}{9} \)[/tex] is calculated as follows:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{\frac{4}{9}} = -\frac{9}{4} \][/tex]
### Step 3: Identify which of the given lines has this slope.
We need to check the slopes of the given lines to find the one with a slope of [tex]\( -\frac{9}{4} \)[/tex].
1. [tex]\( y = -\frac{4}{9}x + \frac{1}{5} \)[/tex]
Slope: [tex]\( -\frac{4}{9} \)[/tex]
2. [tex]\( y = \frac{4}{9}x - \frac{1}{6} \)[/tex]
Slope: [tex]\( \frac{4}{9} \)[/tex]
3. [tex]\( y = -\frac{9}{4}x - \frac{1}{8} \)[/tex]
Slope: [tex]\( -\frac{9}{4} \)[/tex]
4. [tex]\( y = \frac{9}{4}x + \frac{1}{7} \)[/tex]
Slope: [tex]\( \frac{9}{4} \)[/tex]
Comparing these slopes with [tex]\( -\frac{9}{4} \)[/tex], we find that the line:
[tex]\[ y = -\frac{9}{4}x - \frac{1}{8} \][/tex]
has the slope [tex]\( -\frac{9}{4} \)[/tex], which is the negative reciprocal of [tex]\( \frac{4}{9} \)[/tex].
Thus, the line perpendicular to [tex]\( y = \frac{4}{9}x - 7 \)[/tex] is:
[tex]\[ y = -\frac{9}{4}x - \frac{1}{8} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{y = -\frac{9}{4} x - \frac{1}{8}} \][/tex]
