The slope of a line perpendicular to the line whose equation is `(x)/(3)-(y)/(4)=1` is

Options

A. `-3`
B. `-(4)/(3)`
C. `-(3)/(4)`
D. `(1)/(4)`

Explanation

To find the slope of a line perpendicular to the line given by the equation \(\frac{x}{3} - \frac{y}{4} = 1\), we will follow these steps:

### Step 1: Rewrite the equation in slope-intercept form
We start with the equation:
\[ \frac{x}{3} - \frac{y}{4} = 1 \]
To convert this into the slope-intercept form \(y = mx + c\), we first isolate \(y\).

### Step 2: Isolate \(y\)
We can rearrange the equation:
\[ -\frac{y}{4} = 1 - \frac{x}{3} \]
Multiplying through by -1 gives:
\[ \frac{y}{4} = -1 + \frac{x}{3} \]
Now, multiply both sides by 4 to eliminate the fraction:
\[ y = 4\left(-1 + \frac{x}{3}\right) \]
Distributing the 4:
\[ y = -4 + \frac{4x}{3} \]

### Step 3: Identify the slope
Now, we can express the equation in the slope-intercept form:
\[ y = \frac{4}{3}x - 4 \]
From this, we can see that the slope \(m\) of the line is:
\[ m = \frac{4}{3} \]

### Step 4: Find the slope of the perpendicular line
For two lines to be perpendicular, the product of their slopes must equal -1. If \(m_1\) is the slope of the original line and \(m_2\) is the slope of the perpendicular line, we have:
\[ m_1 \cdot m_2 = -1 \]
Substituting \(m_1 = \frac{4}{3}\):
\[ \frac{4}{3} \cdot m_2 = -1 \]
To find \(m_2\), we solve for it:
\[ m_2 = -\frac{3}{4} \]

### Final Answer
The slope of the line perpendicular to the given line is:
\[ \boxed{-\frac{3}{4}} \]