Find the equation of a line that passes through (-2,5) and is perpendicular to the line that passes through (-4,-2) and (5,7)

To find the equation of the line that passes through the point (-2, 5) and is perpendicular to the line passing through (-4, -2) and (5, 7), follow these steps:

### Step 1: Find the slope of the given line
The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the slope formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the given points \((-4, -2)\) and \((5, 7)\) into the formula:

\[
m = \frac{7 - (-2)}{5 - (-4)} = \frac{7 + 2}{5 + 4} = \frac{9}{9} = 1
\]

Thus, the slope of the line passing through (-4, -2) and (5, 7) is \( m = 1 \).

### Step 2: Find the slope of the perpendicular line
The slopes of two perpendicular lines are negative reciprocals of each other. If the slope of the given line is \( m_1 = 1 \), then the slope of the line perpendicular to it is:

\[
m_2 = -\frac{1}{m_1} = -\frac{1}{1} = -1
\]

### Step 3: Use the point-slope form of the line equation
The point-slope form of a line's equation is:

\[
y - y_1 = m(x - x_1)
\]

Where \((x_1, y_1)\) is a point on the line, and \( m \) is the slope. In this case, we have the point (-2, 5) and the slope \( m = -1 \). Substitute these values into the point-slope form:

\[
y - 5 = -1(x - (-2))
\]

\[
y - 5 = -1(x + 2)
\]

### Step 4: Simplify the equation
Distribute the slope on the right-hand side:

\[
y - 5 = -x - 2
\]

Now, add 5 to both sides to get the equation in slope-intercept form:

\[
y = -x - 2 + 5
\]

\[
y = -x + 3
\]

### Final Answer:
The equation of the line that passes through (-2, 5) and is perpendicular to the line passing through (-4, -2) and (5, 7) is:

\[
y = -x + 3
\]

Would you like further details or have any questions?

Here are 5 relative questions to expand on the information:
1. How do you find the equation of a line given two points?
2. What is the meaning of perpendicular slopes in geometry?
3. Can the slope formula be applied to vertical or horizontal lines?
4. What is the difference between slope-intercept form and point-slope form?
5. How do negative reciprocals relate to perpendicular lines in coordinate geometry?

**Tip:** When working with perpendicular lines, always remember their slopes multiply to give \(-1\).

Discover how to find the equation of a line that passes through the point (-2, 5) and is perpendicular to a line passing through (-4, -2) and (5, 7). This problem explores key concepts in algebra and coordinate geometry, including slope calculation and perpendicular lines, suitable for students in Grades 8-10.

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