Consider the line y=(4)/(3)x+8 Find the equation of the line that is parallel to this line and passes through the point (-8,2) Find the equation of the line that is perpendicular to this line and passes through the point (-8,2) Equation of parallel line: Equation of perpendicular line: square

To find the equation of the line that is parallel to the given line $y=\frac{4}{3}x+8$ and passes through the point $(-8,2)$, we need to use the fact that parallel lines have the same slope.<br /><br />Step 1: Identify the slope of the given line.<br />The slope of the given line $y=\frac{4}{3}x+8$ is $\frac{4}{3}$.<br /><br />Step 2: Use the point-slope form of a linear equation to find the equation of the parallel line.<br />The point-slope form of a linear equation is $y-y_1=m(x-x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope.<br /><br />Substituting the values, we have:<br />$y-2=\frac{4}{3}(x+8)$<br /><br />Step 3: Simplify the equation.<br />$y-2=\frac{4}{3}x+\frac{32}{3}$<br />$y=\frac{4}{3}x+\frac{38}{3}$<br /><br />Therefore, the equation of the line that is parallel to the given line and passes through the point $(-8,2)$ is $y=\frac{4}{3}x+\frac{38}{3}$.<br /><br />To find the equation of the line that is perpendicular to the given line and passes through the point $(-8,2)$, we need to use the fact that perpendicular lines have slopes that are negative reciprocals of each other.<br /><br />Step 1: Find the negative reciprocal of the slope of the given line.<br />The negative reciprocal of $\frac{4}{3}$ is $-\frac{3}{4}$.<br /><br />Step 2: Use the point-slope form of a linear equation to find the equation of the perpendicular line.<br />Using the point-slope form again, we have:<br />$y-2=-\frac{3}{4}(x+8)$<br /><br />Step 3: Simplify the equation.<br />$y-2=-\frac{3}{4}x-6$<br />$y=-\frac{3}{4}x-4$<br /><br />Therefore, the equation of the line that is perpendicular to the given line and passes through the point $(-8,2)$ is $y=-\frac{3}{4}x-4$.