To determine which equation represents a line that is perpendicular to the line represented by [tex]\(-\frac{3}{4} x + y = 6\)[/tex], we can follow these steps:
### Step 1: Identify the slope of the given line
First, rearrange the given equation into the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Starting with the given equation:
[tex]\[ -\frac{3}{4} x + y = 6 \][/tex]
we add [tex]\(\frac{3}{4} x\)[/tex] to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3}{4} x + 6 \][/tex]
So, the slope [tex]\( m \)[/tex] of the given line is [tex]\(\frac{3}{4}\)[/tex].
### Step 2: Determine the slope of the perpendicular line
For a line to be perpendicular to another, its slope must be the negative reciprocal of the slope of the original line.
\- The slope of the original line is [tex]\(\frac{3}{4}\)[/tex].
\- The negative reciprocal of [tex]\(\frac{3}{4}\)[/tex] is [tex]\(-\frac{4}{3}\)[/tex].
### Step 3: Identify the correct equation
We need to select the equation from the given options that has a slope of [tex]\(-\frac{4}{3}\)[/tex].
Here are the options with their slopes:
1. [tex]\( y = -\frac{3}{4} x - 2 \)[/tex] [tex]\[ \text{Slope} = -\frac{3}{4} \][/tex]
2. [tex]\( y = -\frac{4}{3} x + 1 \)[/tex] [tex]\[ \text{Slope} = -\frac{4}{3} \][/tex]
3. [tex]\( y = \frac{3}{4} x - 8 \)[/tex] [tex]\[ \text{Slope} = \frac{3}{4} \][/tex]
4. [tex]\( y = \frac{4}{3} x + 6 \)[/tex] [tex]\[ \text{Slope} = \frac{4}{3} \][/tex]
### Step 4: Conclude the correct option
The equation [tex]\( y = -\frac{4}{3} x + 1 \)[/tex] has the slope [tex]\(-\frac{4}{3}\)[/tex], which is the negative reciprocal of the slope of the given line. Therefore, this is the correct equation representing a line that is perpendicular to the given line.
### Final Answer
The correct equation representing a line that is perpendicular to [tex]\(-\frac{3}{4} x + y = 6\)[/tex] is:
[tex]\[ y = -\frac{4}{3} x + 1 \][/tex]
