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To solve the problem, we need to find the equation of a line in slope-intercept form (i.e., \( y = mx + b \)) that:

1. Passes through the point \( (2, -4) \).
2. Is perpendicular to the line \( y = \frac{2}{3}x + 5 \).

### Step 1: Find the slope of the given line.
The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope. In this case, the slope of the given line \( y = \frac{2}{3}x + 5 \) is \( \frac{2}{3} \).

### Step 2: Find the slope of the perpendicular line.
The slope of two perpendicular lines is the negative reciprocal of each other. The negative reciprocal of \( \frac{2}{3} \) is \( -\frac{3}{2} \).

So, the slope of the line we are trying to find is \( -\frac{3}{2} \).

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

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

where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. We know that \( m = -\frac{3}{2} \) and \( (x_1, y_1) = (2, -4) \).

Substitute these values into the point-slope form:

\[
y - (-4) = -\frac{3}{2}(x - 2)
\]

\[
y + 4 = -\frac{3}{2}(x - 2)
\]

### Step 4: Simplify to slope-intercept form.
Distribute \( -\frac{3}{2} \) on the right-hand side:

\[
y + 4 = -\frac{3}{2}x + 3
\]

Now, subtract 4 from both sides to solve for \( y \):

\[
y = -\frac{3}{2}x + 3 - 4
\]

\[
y = -\frac{3}{2}x - 1
\]

### Step 5: Compare with the answer choices.
The correct equation is \( y = -\frac{3}{2}x - 1 \), which matches option **B**.

Would you like further details or explanations? Here are some related questions:

1. How do you calculate the slope of a line given two points?
2. What is the general method for finding the equation of a line parallel to a given line?
3. How can you verify if two lines are perpendicular algebraically?
4. What is the difference between point-slope form and slope-intercept form?
5. How does the y-intercept change when transforming a line equation?

**Tip**: When finding the equation of a line perpendicular to another, always remember to use the negative reciprocal of the original line's slope.

What is the equation in slope-intercept form of the line that passes through the point (2, −4) and is perpendicular to the line represented by y = (2/3)x + 5?

Solve for the equation of a line in slope-intercept form that passes through the point (2, -4) and is perpendicular to y = (2/3)x + 5. Includes key algebraic concepts such as slopes, perpendicular lines, and using point-slope form. Suitable for students in Grades 8-10.

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