Find the equation of the line that is parallel to this line and passes through the point \( (5,4) \). Find the equation of the line that is perpendicular to this line and passes through the point \( (5,4) \).

To find the equations of the lines that are parallel and perpendicular to a given line, we first need the equation of the original line. However, since the equation of the line is not provided in your question, I will demonstrate the process using a general line equation in slope-intercept form, \( y = mx + b \), where \( m \) is the slope. ### Step 1: Find the slope of the given line Assume the equation of the given line is \( y = mx + b \). The slope of this line is \( m \). ### Step 2: Find the equation of the parallel line A line that is parallel to another line has the same slope. Therefore, the slope of the parallel line will also be \( m \). Using the point-slope form of the equation of a line, which is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point through which the line passes, we can substitute \( (5, 4) \) for \( (x_1, y_1) \): \[ y - 4 = m(x - 5) \] This can be rearranged to the slope-intercept form: \[ y = m(x - 5) + 4 \] \[ y = mx - 5m + 4 \] ### Step 3: Find the equation of the perpendicular line A line that is perpendicular to another line has a slope that is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line will be \( -\frac{1}{m} \). Using the point-slope form again: \[ y - 4 = -\frac{1}{m}(x - 5) \] This can be rearranged to: \[ y - 4 = -\frac{1}{m}x + \frac{5}{m} \] \[ y = -\frac{1}{m}x + \frac{5}{m} + 4 \] ### Summary 1. The equation of the line parallel to the given line and passing through \( (5, 4) \) is: \[ y = mx - 5m + 4 \] 2. The equation of the line perpendicular to the given line and passing through \( (5, 4) \) is: \[ y = -\frac{1}{m}x + \frac{5}{m} + 4 \] To provide specific equations, please provide the equation of the original line.