Write the equation of the line that passes through the given point and is perpendicular to the given line. (-6, -3); y = -2/5x

To find the equation of a line that passes through the point \((-6, -3)\) and is perpendicular to the line \(y = -\frac{2}{5}x\), follow these steps:

### Step 1: Identify the slope of the given line
The equation of the given line is in slope-intercept form \(y = mx + b\), where \(m\) is the slope. From the equation \(y = -\frac{2}{5}x\), the slope is:
\[
m_1 = -\frac{2}{5}
\]

### Step 2: Find the slope of the perpendicular line
The slope of two perpendicular lines is related by the negative reciprocal. If \(m_1 = -\frac{2}{5}\), then the slope of the perpendicular line \(m_2\) is:
\[
m_2 = \frac{5}{2}
\]
This is the negative reciprocal of \(-\frac{2}{5}\).

### Step 3: Use the point-slope form of the line equation
The point-slope form of a line is:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point through which the line passes, and \(m\) is the slope of the line.

We know that the line passes through \((-6, -3)\) and has a slope of \(\frac{5}{2}\). Substituting these values into the point-slope form:
\[
y - (-3) = \frac{5}{2}(x - (-6))
\]
Simplifying:
\[
y + 3 = \frac{5}{2}(x + 6)
\]

### Step 4: Simplify to slope-intercept form
Now, expand and simplify the equation:
\[
y + 3 = \frac{5}{2}(x + 6)
\]
\[
y + 3 = \frac{5}{2}x + \frac{5}{2} \times 6
\]
\[
y + 3 = \frac{5}{2}x + 15
\]
Now, subtract 3 from both sides:
\[
y = \frac{5}{2}x + 12
\]

### Final Answer:
The equation of the line that passes through \((-6, -3)\) and is perpendicular to the line \(y = -\frac{2}{5}x\) is:
\[
y = \frac{5}{2}x + 12
\]

Would you like further details or have any questions?

Here are 5 relative questions to expand your understanding:
1. How do you find the slope of a line given two points?
2. What is the significance of the slope in a linear equation?
3. Can you derive the slope-intercept form from the point-slope form?
4. How do you know two lines are perpendicular based on their slopes?
5. What is the general form of a linear equation, and how does it differ from slope-intercept form?

**Tip:** The slope of two perpendicular lines is always the negative reciprocal of each other!

Learn how to find the equation of a line that passes through the point (-6, -3) and is perpendicular to the line y = -2/5x. This problem uses the point-slope form and covers key concepts like slopes of perpendicular lines. Suitable for students in Grades 9-12.

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