Question

Write an equation of the line that passes through (3,2) and is perpendicular to the line defined by x-4y= -1. Write the answer in slope-Intercept form (If possible) and in standard form (Ax + By=C) with smallest Integer coefficients. Use the "Cannot be written" button, if applicable.

Answer 1

To find the equation of the line that passes through the point (3, 2) and is perpendicular to the line defined by the equation \( x - 4y = -1 \), we first need to determine the slope of the given line.

1. **Convert to slope-intercept form (y = mx + b)**:
\[
x - 4y = -1
\]
Rearranging gives:
\[
-4y = -x - 1 \quad \Rightarrow \quad y = \frac{1}{4}x + \frac{1}{4}
\]
Here, the slope \( m \) of the line is \( \frac{1}{4} \).

2. **Find the slope of the perpendicular line**:
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Given the slope of the original line is \( \frac{1}{4} \), the slope of the perpendicular line will be:
\[
m_{perpendicular} = -\frac{1}{\left(\frac{1}{4}\right)} = -4
\]

3. **Use point-slope form to find the equation**:
We have the slope and a point through which the line passes, which is (3, 2). The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Substituting in the values:
\[
y - 2 = -4(x - 3)
\]
Expanding this gives:
\[
y - 2 = -4x + 12 \quad \Rightarrow \quad y = -4x + 14
\]

4. **Convert to standard form**:
The slope-intercept form of the line is \( y = -4x + 14 \). To convert it to standard form \( Ax + By = C \):
\[
4x + y = 14
\]
Here, \( A = 4 \), \( B = 1 \), and \( C = 14 \).

**Final Results**:
- **Slope-intercept form**: \( y = -4x + 14 \)
- **Standard form**: \( 4x + y = 14 \)

Both forms are correct and use the smallest integer coefficients.