Find equations of the lines that pass through the given point and are parallel to and perpendicular to the given line. 2x+3y=5, (-(1)/(2),(5)/(3)) (a) parallel to the given line square (b) perpendicular ar to the given line square

To find the equations of the lines that pass through the given point and are parallel to and perpendicular to the given line, we need to follow these steps:<br /><br />1. Find the slope of the given line.<br />2. Use the point-slope form of a linear equation to find the equation of the line parallel to the given line.<br />3. Find the slope of the line perpendicular to the given line.<br />4. Use the point-slope form of a linear equation to find the equation of the line perpendicular to the given line.<br /><br />Let's start with step 1:<br /><br />1. Find the slope of the given line:<br />The given line is in the form of Ax + By = C, where A = 2, B = 3, and C = 5. To find the slope, we need to rewrite the equation in the slope-intercept form, y = mx + b, where m is the slope.<br /><br />Rearranging the equation, we get:<br />3y = -2x + 5<br />y = (-2/3)x + 5/3<br /><br />The slope of the given line is -2/3.<br /><br />Now, let's move on to step 2:<br /><br />2. Find the equation of the line parallel to the given line:<br />Since the line parallel to the given line will have the same slope, the slope of the parallel line is also -2/3.<br /><br />Using the point-slope form of a linear equation, we have:<br />y - y1 = m(x - x1)<br /><br />Substituting the given point (-1/2, 5/3) and the slope -2/3, we get:<br />y - 5/3 = (-2/3)(x + 1/2)<br /><br />Simplifying the equation, we have:<br />y - 5/3 = (-2/3)x - 1/3<br />y = (-2/3)x + 4/3<br /><br />Therefore, the equation of the line parallel to the given line is:<br />y = (-2/3)x + 4/3<br /><br />Now, let's move on to step 3:<br /><br />3. Find the slope of the line perpendicular to the given line:<br />The slope of the line perpendicular to the given line is the negative reciprocal of the slope of the given line.<br /><br />The slope of the given line is -2/3, so the slope of the perpendicular line is 3/2.<br /><br />Finally, let's move on to step 4:<br /><br />4. Find the equation of the line perpendicular to the given line:<br />Using the point-slope form of a linear equation, we have:<br />y - y1 = m(x - x1)<br /><br />Substituting the given point (-1/2, 5/3) and the slope 3/2, we get:<br />y - 5/3 = (3/2)(x + 1/2)<br /><br />Simplifying the equation, we have:<br />y - 5/3 = (3/2)x + 3/4<br />y = (3/2)x + 23/12<br /><br />Therefore, the equation of the line perpendicular to the given line is:<br />y = (3/2)x + 23/12<br /><br />In summary, the equations of the lines that pass through the given point and are parallel to and perpendicular to the given line are:<br />(a) parallel to the given line: y = (-2/3)x + 4/3<br />(b) perpendicular to the given line: y = (3/2)x + 23/12