What is an equation of the line that passes through the point left parenthesis, 4, comma, minus, 6, right parenthesis(4,−6) and is perpendicular to the line 2, x, minus, 3, y, equals, 272*x*−3*y*=27?

To find the equation of the line that passes through the point \((4, -6)\) and is perpendicular to the line \(2x - 3y = 27\), we need to follow these steps:

### Step 1: Find the slope of the given line
The equation of the given line is \(2x - 3y = 27\). We can rewrite this equation in the slope-intercept form \(y = mx + b\) to find the slope.

Start by solving for \(y\):
\[
2x - 3y = 27
\]
\[
-3y = -2x + 27
\]
\[
y = \frac{2}{3}x - 9
\]
The slope \(m_1\) of the given line is \(\frac{2}{3}\).

### Step 2: Find the slope of the perpendicular line
The slope \(m_2\) of the line perpendicular to the given line is the negative reciprocal of \(m_1\). Thus:
\[
m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2}
\]

### Step 3: Use the point-slope form of the equation
We now know the slope \(m_2 = -\frac{3}{2}\) and that the line passes through the point \((4, -6)\). We can use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Substitute \(m = -\frac{3}{2}\), \(x_1 = 4\), and \(y_1 = -6\):
\[
y - (-6) = -\frac{3}{2}(x - 4)
\]
\[
y + 6 = -\frac{3}{2}(x - 4)
\]

### Step 4: Simplify to get the equation
Distribute the slope on the right-hand side:
\[
y + 6 = -\frac{3}{2}x + 6
\]
Subtract 6 from both sides:
\[
y = -\frac{3}{2}x
\]

So, the equation of the line that passes through \((4, -6)\) and is perpendicular to the line \(2x - 3y = 27\) is:
\[
\boxed{y = -\frac{3}{2}x}
\]

Would you like to go over any details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you find the equation of a line that is parallel to a given line?
2. What is the general form of the equation of a line?
3. How do you convert a line equation from standard form to slope-intercept form?
4. How do you determine if two lines are perpendicular by comparing their slopes?
5. What happens to the equation of a line when it is translated horizontally or vertically?

**Tip:** When working with perpendicular lines, remember that their slopes are always negative reciprocals of each other.

Learn how to find the equation of a line that passes through the point (4, -6) and is perpendicular to the line 2x - 3y = 27. This problem involves using the point-slope form and understanding perpendicular slopes in geometry. Suitable for Grades 9-12 students.

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