What is the slope of the line perpendicular to the line `(x)/(4) + (y)/(3) = 1` ?

Options

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

Answer

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

### Step 1: Convert the equation to slope-intercept form
Start with the equation:
\[ \frac{x}{4} + \frac{y}{3} = 1 \]
To eliminate the fractions, find a common denominator. The least common multiple of 4 and 3 is 12. Multiply the entire equation by 12:
\[ 12 \left(\frac{x}{4}\right) + 12 \left(\frac{y}{3}\right) = 12 \cdot 1 \]
This simplifies to:
\[ 3x + 4y = 12 \]

### Step 2: Rearrange to solve for \(y\)
Now, rearrange the equation to isolate \(y\):
\[ 4y = -3x + 12 \]
Divide every term by 4:
\[ y = -\frac{3}{4}x + 3 \]

### Step 3: Identify the slope of the original line
From the equation \(y = -\frac{3}{4}x + 3\), we can see that the slope \(m_1\) of the original line is:
\[ m_1 = -\frac{3}{4} \]

### Step 4: Use the property of perpendicular slopes
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, if \(m_2\) is the slope of the perpendicular line, we have:
\[ m_1 \cdot m_2 = -1 \]
Substituting \(m_1\):
\[ -\frac{3}{4} \cdot m_2 = -1 \]

### Step 5: Solve for \(m_2\)
To find \(m_2\), divide both sides by \(-\frac{3}{4}\):
\[ m_2 = \frac{-1}{-\frac{3}{4}} = \frac{4}{3} \]

### Conclusion
Thus, the slope of the line that is perpendicular to the given line is:
\[ \boxed{\frac{4}{3}} \]