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 find the equation of the line in slope-intercept form \(y = mx + b\), we need the slope \(m\) of the line and a point it passes through.

### Step 1: Find the slope of the perpendicular line
The given line is \(y = \frac{2}{3}x + 5\), where the slope is \(m_1 = \frac{2}{3}\).

For a line to be perpendicular to this one, its slope must be the negative reciprocal of \(m_1\). The negative reciprocal of \(\frac{2}{3}\) is \(-\frac{3}{2}\). Therefore, the slope \(m_2\) of the perpendicular line is:

\[
m_2 = -\frac{3}{2}
\]

### Step 2: Use the point-slope form
The perpendicular line passes through the point \((2, -4)\). Using the point-slope form of a line equation:

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

where \((x_1, y_1)\) is the point \((2, -4)\) and \(m = -\frac{3}{2}\):

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

Simplify:

\[
y + 4 = -\frac{3}{2}(x - 2)
\]
\[
y + 4 = -\frac{3}{2}x + 3
\]
\[
y = -\frac{3}{2}x + 3 - 4
\]
\[
y = -\frac{3}{2}x - 1
\]

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

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

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do you find the equation of a line given two points?
2. What is the relationship between the slopes of perpendicular lines?
3. How do you convert from point-slope form to slope-intercept form?
4. What is the significance of the y-intercept in the slope-intercept form?
5. How do you find the equation of a line parallel to a given line?

**Tip:** Always remember, perpendicular lines have slopes that 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 perpendicular to y = 2/3x + 5 that passes through the point (2, -4). This problem covers key algebraic concepts such as the slope-intercept form and negative reciprocal slopes. Suitable for Grades 8-10.

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