Find the slope of the line which makes an angle of `45^(@)` with a line of slope `-6/(5)`.

Answer

To find the slope of the line that makes an angle of \( 45^\circ \) with a line of slope \( m_1 = -\frac{6}{5} \), we can use the formula for the tangent of the angle between two lines. The formula is:

\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]

where:
- \( \theta \) is the angle between the two lines,
- \( m_1 \) is the slope of the first line,
- \( m_2 \) is the slope of the second line.

### Step 1: Identify the known values
We know:
- \( \theta = 45^\circ \)
- \( m_1 = -\frac{6}{5} \)

### Step 2: Substitute the known values into the formula
Since \( \tan 45^\circ = 1 \), we can substitute into the formula:

\[ 1 = \frac{-\frac{6}{5} - m_2}{1 + \left(-\frac{6}{5}\right) m_2} \]

### Step 3: Simplify the equation
Multiply both sides by the denominator to eliminate the fraction:

\[ 1 \cdot \left(1 + \left(-\frac{6}{5}\right) m_2\right) = -\frac{6}{5} - m_2 \]

This simplifies to:

\[ 1 - \frac{6}{5} m_2 = -\frac{6}{5} - m_2 \]

### Step 4: Rearrange the equation
Rearranging gives:

\[ 1 + \frac{6}{5} + m_2 = \frac{6}{5} m_2 \]

Combine like terms:

\[ \frac{11}{5} = \frac{6}{5} m_2 + m_2 \]

### Step 5: Combine the terms involving \( m_2 \)
This can be rewritten as:

\[ \frac{11}{5} = \left(\frac{6}{5} + 1\right) m_2 \]

Convert 1 to a fraction with a common denominator:

\[ 1 = \frac{5}{5} \]

Thus:

\[ \frac{11}{5} = \left(\frac{6}{5} + \frac{5}{5}\right) m_2 = \frac{11}{5} m_2 \]

### Step 6: Solve for \( m_2 \)
Now, we can solve for \( m_2 \):

\[ m_2 = \frac{11/5}{11/5} = 1 \]

### Step 7: Final result
Thus, the slope \( m_2 \) of the line that makes an angle of \( 45^\circ \) with the line of slope \( -\frac{6}{5} \) is:

\[ m_2 = -1 \]