Find the equation of the line that passes through ​(minus2​,5​) and is perpendicular to the line
passing through left parenthesis negative 6 comma one half right parenthesis
 and left parenthesis negative 4 comma two thirds right parenthesis
. Write the equation in​ slope-intercept form.

To find the equation of the line that passes through the point \((-2, 5)\) and is perpendicular to the line passing through the points \((-6, \frac{1}{2})\) and \((-4, \frac{2}{3})\), 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 as:

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

Using the points \((-6, \frac{1}{2})\) and \((-4, \frac{2}{3})\):

\[
m = \frac{\frac{2}{3} - \frac{1}{2}}{-4 - (-6)} = \frac{\frac{2}{3} - \frac{1}{2}}{2}
\]

First, find \(\frac{2}{3} - \frac{1}{2}\). The common denominator is 6:

\[
\frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6}
\]

\[
\frac{2}{3} - \frac{1}{2} = \frac{4}{6} - \frac{3}{6} = \frac{1}{6}
\]

So:

\[
m = \frac{\frac{1}{6}}{2} = \frac{1}{12}
\]

### Step 2: Find the slope of the line perpendicular to the given line.
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of \(\frac{1}{12}\) is:

\[
-\frac{12}{1} = -12
\]

### Step 3: Use the point-slope form to find the equation of the line.
The point-slope form of a line equation is:

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

Substituting \(m = -12\) and the point \((-2, 5)\):

\[
y - 5 = -12(x + 2)
\]

### Step 4: Simplify to get the slope-intercept form.
Distribute the \(-12\) and solve for \(y\):

\[
y - 5 = -12x - 24
\]

\[
y = -12x - 19
\]

### Final Answer:
The equation of the line in slope-intercept form is:

\[
\boxed{y = -12x - 19}
\]

Would you like further details or have any questions?

### Follow-up Questions
1. What is the slope-intercept form of a line?
2. How do you find the slope of a line given two points?
3. What is the significance of the negative reciprocal in perpendicular lines?
4. Can you explain how to find the common denominator when subtracting fractions?
5. How would the equation change if the line were parallel instead of perpendicular?

### Tip
Remember, when working with perpendicular lines, always take the negative reciprocal of the original slope to find the slope of the perpendicular line.

Find the equation of the line that passes through (-2, 5) and is perpendicular to the line passing through (-6, 1/2) and (-4, 2/3). Write the equation in slope-intercept form.

Learn how to find the equation of a line that passes through the point (-2, 5) and is perpendicular to a given line. This detailed solution covers essential algebraic concepts such as slopes, point-slope form, and slope-intercept form. Perfect for students in Grades 9-10.

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