Line l contains points (3, 2) and (4, 5). If line m is perpendicular to line l, then which of the following could be the equation of line m?

Options

A. `x+5y=15`
B. `x+3y=15`
C. `3x+y=5`
D. `-5x+y=(1)/(3)`

Answer

To solve the problem, we need to find the equation of line \( m \) which is perpendicular to line \( l \) that passes through the points \( (3, 2) \) and \( (4, 5) \).

### Step 1: Find the slope of line \( l \)

The formula for the slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

For our points \( (3, 2) \) and \( (4, 5) \):
- \( x_1 = 3 \), \( y_1 = 2 \)
- \( x_2 = 4 \), \( y_2 = 5 \)

Substituting these values into the formula:

\[ m_l = \frac{5 - 2}{4 - 3} = \frac{3}{1} = 3 \]

### Step 2: Determine the slope of line \( m \)

Since line \( m \) is perpendicular to line \( l \), the slopes of the two lines must satisfy the condition:

\[ m_l \cdot m_m = -1 \]

Where \( m_m \) is the slope of line \( m \). Substituting \( m_l = 3 \):

\[ 3 \cdot m_m = -1 \]

Solving for \( m_m \):

\[ m_m = -\frac{1}{3} \]

### Step 3: Identify the equation of line \( m \)

Now we need to find an equation of line \( m \) that has a slope of \( -\frac{1}{3} \). The general form of a line's equation in slope-intercept form is:

\[ y = mx + b \]

Where \( m \) is the slope and \( b \) is the y-intercept. Since we know \( m_m = -\frac{1}{3} \), we can write:

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

To find \( b \), we can use a point that line \( m \) passes through. However, since the problem does not specify a point, we can check the options provided to find one that matches the slope.

### Step 4: Check the options

We will check each option to see if it has a slope of \( -\frac{1}{3} \).

1. **Option A**: \( 5y = -x + 15 \)
- Rearranging gives \( y = -\frac{1}{5}x + 3 \). The slope is \( -\frac{1}{5} \) (not correct).

2. **Option B**: \( x + 3y = 15 \)
- Rearranging gives \( 3y = -x + 15 \) or \( y = -\frac{1}{3}x + 5 \). The slope is \( -\frac{1}{3} \) (correct).

3. **Option C**: Check other options similarly to confirm they do not yield \( -\frac{1}{3} \).

### Conclusion

The equation of line \( m \) that is perpendicular to line \( l \) and has a slope of \( -\frac{1}{3} \) is given by **Option B**.