The line through A(-2, 3) and B(4, b) is perpendicular to the line `2x - 4y = 5`. Find the value of b.

Answer

To find the value of \( b \) such that the line through points \( A(-2, 3) \) and \( B(4, b) \) is perpendicular to the line given by the equation \( 2x - 4y = 5 \), we can follow these steps:

### Step 1: Find the slope of the line \( 2x - 4y = 5 \)

First, we need to convert the equation into slope-intercept form \( y = mx + c \), where \( m \) is the slope.

Starting with the equation:
\[ 2x - 4y = 5 \]
Rearranging it gives:
\[ -4y = -2x + 5 \]
Dividing through by -4:
\[ y = \frac{1}{2}x - \frac{5}{4} \]
Thus, the slope \( m_L \) of the line is:
\[ m_L = \frac{1}{2} \]

**Hint:** To find the slope from the standard form, rearrange the equation to isolate \( y \).

### Step 2: Use the perpendicular slope relationship

The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, if \( m_L = \frac{1}{2} \), then the slope \( m_{AB} \) of line \( AB \) is:
\[ m_{AB} = -\frac{1}{m_L} = -2 \]

**Hint:** Remember that for two perpendicular lines, the product of their slopes equals -1.

### Step 3: Calculate the slope of line \( AB \)

Using the coordinates of points \( A(-2, 3) \) and \( B(4, b) \), the slope \( m_{AB} \) can be calculated as:
\[ m_{AB} = \frac{b - 3}{4 - (-2)} = \frac{b - 3}{4 + 2} = \frac{b - 3}{6} \]

**Hint:** The slope formula is \( \frac{y_2 - y_1}{x_2 - x_1} \).

### Step 4: Set the slopes equal to each other

Since we have found that \( m_{AB} = -2 \), we can set up the equation:
\[ \frac{b - 3}{6} = -2 \]

**Hint:** Set the calculated slope equal to the perpendicular slope.

### Step 5: Solve for \( b \)

Now, we will solve the equation:
\[ b - 3 = -2 \times 6 \]
\[ b - 3 = -12 \]
Adding 3 to both sides:
\[ b = -12 + 3 \]
\[ b = -9 \]

**Hint:** Isolate \( b \) by performing inverse operations.

### Final Answer

The value of \( b \) is:
\[ \boxed{-9} \]