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=23x+5
.
A.y=−32x+5

B.y=−32x−1

C.y=23x−4

D.y=23x+2


To solve this, we need to find the equation of the line in slope-intercept form (i.e., \(y = mx + b\)) that passes through the point \((2, -4)\) and is perpendicular to the line \(y = \frac{2}{3}x + 5\).

### Step 1: Determine the slope of the perpendicular line
- The slope of the given line is \(\frac{2}{3}\).
- The slopes of perpendicular lines are negative reciprocals of each other.
  
So, the slope of the line perpendicular to \(y = \frac{2}{3}x + 5\) will be:
\[
m_{\text{perpendicular}} = -\frac{3}{2}
\]

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

### Step 3: Simplify to slope-intercept form
Distribute the slope \(-\frac{3}{2}\):
\[
y + 4 = -\frac{3}{2}x + 3
\]
Now, subtract 4 from both sides to isolate \(y\):
\[
y = -\frac{3}{2}x + 3 - 4
\]
\[
y = -\frac{3}{2}x - 1
\]

### Conclusion:
The equation of the line is \(y = -\frac{3}{2}x - 1\), which corresponds to option **B**.

### Answer: 
**B. \(y = -\frac{3}{2}x - 1\)**

Do you need any further details or clarifications?

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Here are 5 related questions:
1. What is the slope of a line perpendicular to a line with slope 4?
2. How do you find the equation of a line parallel to another given line?
3. How would you convert a line equation from point-slope form to standard form?
4. How do you find the slope of a line given two points?
5. What is the difference between parallel and perpendicular lines?

**Tip:** Remember that perpendicular slopes are negative reciprocals of each other!

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=2/3x+5?

Learn how to find the equation of a line in slope-intercept form that passes through the point (2, -4) and is perpendicular to the line y=2/3x+5. This problem involves key algebraic concepts such as slopes of perpendicular lines and point-slope form, suitable for students in Grades 9-10.

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