use the given conditions to write an equation for the line in point-slope form and in slope intercept form. passing through (4,-7) and perpendicular to the line whose equation is y = 1/4x + 3

To write the equation of a line in both point-slope form and slope-intercept form that is perpendicular to a given line and passes through a specific point, we can follow these steps:

### Step 1: Determine the Slope of the Perpendicular Line

The given line is \(y = \frac{1}{4}x + 3\). The slope of this line is \(\frac{1}{4}\).

Lines that are perpendicular to each other have slopes that are negative reciprocals. Therefore, the slope \(m\) of the line perpendicular to the given line is:
\[
m = -\frac{1}{\left(\frac{1}{4}\right)} = -4
\]

### Step 2: Use the Point-Slope Form

The point-slope form of the equation of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.

Given point: \((4, -7)\)  
Slope: \(-4\)

Substituting these values into the point-slope form:
\[
y - (-7) = -4(x - 4)
\]
\[
y + 7 = -4(x - 4)
\]

This is the equation in **point-slope form**.

### Step 3: Convert to Slope-Intercept Form

To convert the equation to slope-intercept form \(y = mx + b\), we solve for \(y\):
\[
y + 7 = -4(x - 4)
\]
Distribute the \(-4\):
\[
y + 7 = -4x + 16
\]
Subtract 7 from both sides to isolate \(y\):
\[
y = -4x + 16 - 7
\]
\[
y = -4x + 9
\]

This is the equation in **slope-intercept form**.

### Summary

- **Point-Slope Form**: \(y + 7 = -4(x - 4)\)
- **Slope-Intercept Form**: \(y = -4x + 9\)

These equations describe the line that passes through the point \((4, -7)\) and is perpendicular to the line \(y = \frac{1}{4}x + 3\).

---

Do you want details or have any questions about these equations? Here are some related questions for further practice:

1. What is the equation of a line perpendicular to \(y = 2x - 5\) passing through the origin?
2. How do you find the slope of a line parallel to \(y = -3x + 2\)?
3. Given the point (6, 2) and slope \(-\frac{1}{2}\), what is the equation in point-slope form?
4. Convert the point-slope form \(y - 1 = \frac{2}{3}(x - 5)\) to slope-intercept form.
5. What is the slope-intercept form of a line passing through \((0, -3)\) with slope \(\frac{5}{2}\)?
6. How do you determine if two lines are perpendicular just by looking at their equations?
7. Find the slope-intercept form of the line parallel to \(y = 3x + 7\) and passing through \((-2, 4)\).
8. If a line is vertical, what is its slope and how is its equation written?

**Tip:** To quickly determine if two lines are perpendicular, check if the product of their slopes is \(-1\).

Learn how to find the equation of a line perpendicular to y = 1/4x + 3 passing through the point (4, -7). Includes step-by-step solutions in both point-slope and slope-intercept forms. Suitable for students in Grades 7-9 studying algebra and linear equations.

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