Question

Use the given conditions to write an equation for the line in​ point-slope form and in​ slope-intercept form.
Passing through ​(4​,minus5​) and perpendicular to the line whose equation is y equals one fifth x plus 3
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Part 1
Write an equation for the line in​ point-slope form.
  
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​(Simplify your answer. Use integers or fractions for any numbers in the​ equation.)

Answer 1

To find the equation of the line in point-slope form that passes through the point \( (4, -5) \) and is perpendicular to the line given by the equation \( y = \frac{1}{5}x + 3 \), we can follow these steps:

1. **Determine the slope of the original line**: The slope of the line \( y = \frac{1}{5}x + 3 \) is \( \frac{1}{5} \).

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. The negative reciprocal of \( \frac{1}{5} \) is \( -5 \).

3. **Use the point-slope form**: Point-slope form of a line is given by the formula:
\[
y - y_1 = m(x - x_1)
\]
Where \( (x_1, y_1) \) is a point on the line (which is \( (4, -5) \)) and \( m \) is the slope of the line (which is \( -5 \)).

Substituting into the point-slope form:
\[
y - (-5) = -5(x - 4)
\]

This simplifies to:
\[
y + 5 = -5(x - 4)
\]

So, the equation of the line in point-slope form is:
\[
y + 5 = -5(x - 4)
\]

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### Part 2: Slope-Intercept Form

Now, let's convert the point-slope form to slope-intercept form \( (y = mx + b) \).

Starting from the point-slope form:
\[
y + 5 = -5(x - 4)
\]

Distribute the \(-5\) on the right side:
\[
y + 5 = -5x + 20
\]

Now, subtract \( 5 \) from both sides:
\[
y = -5x + 20 - 5
\]
\[
y = -5x + 15
\]

The equation of the line in slope-intercept form is:
\[
y = -5x + 15
\]

### Summary:
- **Point-slope form**: \( y + 5 = -5(x - 4) \)
- **Slope-intercept form**: \( y = -5x + 15 \)