Find the equation of a straight line perpendicular to the straight line 3x + 4y = 7 and passing through the point (3, -3).

Options

A. x - 3y = 21
B. 4x = 3y
C. 4 x - 3y = 21
D. 3x - 4y = 21

Answer

To find the equation of a straight line that is perpendicular to the line given by the equation \(3x + 4y = 7\) and passes through the point \((3, -3)\), we can follow these steps:

### Step 1: Find the slope of the given line
First, we need to rewrite the equation \(3x + 4y = 7\) in slope-intercept form \(y = mx + b\), where \(m\) is the slope.

1. Rearranging the equation:
\[ 4y = -3x + 7 \]
2. Dividing by 4:
\[ y = -\frac{3}{4}x + \frac{7}{4} \]

The slope \(m_1\) of the given line is \(-\frac{3}{4}\).

### Step 2: Find the slope of the perpendicular line
For two lines to be perpendicular, the product of their slopes must equal \(-1\). If the slope of the given line is \(m_1\), then the slope \(m\) of the perpendicular line can be found using the formula:
\[ m \cdot m_1 = -1 \]
Substituting \(m_1 = -\frac{3}{4}\):
\[ m \cdot \left(-\frac{3}{4}\right) = -1 \]
Solving for \(m\):
\[ m = \frac{4}{3} \]

### Step 3: Use the point-slope form to find the equation of the perpendicular line
Now that we have the slope of the perpendicular line, we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Where \((x_1, y_1)\) is the point through which the line passes, which is \((3, -3)\).

Substituting the values:
\[ y - (-3) = \frac{4}{3}(x - 3) \]
This simplifies to:
\[ y + 3 = \frac{4}{3}(x - 3) \]

### Step 4: Simplify the equation
Now, we will simplify the equation:
1. Distributing \(\frac{4}{3}\):
\[ y + 3 = \frac{4}{3}x - 4 \]
2. Subtracting 3 from both sides:
\[ y = \frac{4}{3}x - 4 - 3 \]
\[ y = \frac{4}{3}x - 7 \]

### Final Equation
The equation of the line that is perpendicular to \(3x + 4y = 7\) and passes through the point \((3, -3)\) is:
\[ y = \frac{4}{3}x - 7 \]