Question: Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form.
Passing through $(5, -4)$ and perpendicular to the line whose equation is $y = \frac{1}{5} x + 4$
Write an equation for the line in point-slope form.
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Write an equation for the line in slope-intercept form.
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
1. **State the problem:**
We need to find the equation of a line passing through the point $(5, -4)$ and perpendicular to the line $y = \frac{1}{5} x + 4$.
2. **Find the slope of the given line:**
The given line is in slope-intercept form $y = mx + b$ where $m = \frac{1}{5}$.
3. **Find the slope of the perpendicular line:**
The slope of a line perpendicular to another is the negative reciprocal of the original slope.
$$m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{1}{5}} = -5$$
4. **Write the point-slope form:**
The point-slope form is:
$$y - y_1 = m(x - x_1)$$
Using point $(5, -4)$ and slope $-5$:
$$y - (-4) = -5(x - 5)$$
Simplify:
$$y + 4 = -5(x - 5)$$
5. **Write the slope-intercept form:**
Start from point-slope form:
$$y + 4 = -5x + 25$$
Subtract 4 from both sides:
$$y = -5x + 25 - 4$$
$$y = -5x + 21$$
**Final answers:**
- Point-slope form: $$y + 4 = -5(x - 5)$$
- Slope-intercept form: $$y = -5x + 21$$