Consider the line \( y=\frac{5}{4} x+2 \). Find the equation of the line that is parallel to this line and passes through the point \( (-4,-4) \). Find the equation of the line that is perpendicular to this line and passes through the point \( (-4,-4) \).
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To find the equation of the line parallel to \( y = \frac{5}{4}x + 2 \) that passes through the point \( (-4, -4) \), we keep the same slope, which is \( \frac{5}{4} \). Using the point-slope form \( y - y_1 = m(x - x_1) \): \[ y - (-4) = \frac{5}{4}(x - (-4)) \] This simplifies to: \[ y + 4 = \frac{5}{4}(x + 4) \] \[ y + 4 = \frac{5}{4}x + 5 \] \[ y = \frac{5}{4}x + 1 \] Now, for the equation of the line perpendicular to \( y = \frac{5}{4}x + 2 \), we need to find the negative reciprocal of the slope. The slope of the perpendicular line is \( -\frac{4}{5} \). Using the point-slope form again: \[ y - (-4) = -\frac{4}{5}(x - (-4)) \] This simplifies to: \[ y + 4 = -\frac{4}{5}(x + 4) \] \[ y + 4 = -\frac{4}{5}x - \frac{16}{5} \] \[ y = -\frac{4}{5}x - \frac{16}{5} - 4 \] \[ y = -\frac{4}{5}x - \frac{16}{5} - \frac{20}{5} \] \[ y = -\frac{4}{5}x - \frac{36}{5} \] So, the equations are: - Parallel line: \( y = \frac{5}{4}x + 1 \) - Perpendicular line: \( y = -\frac{4}{5}x - \frac{36}{5} \)