Mathematics
Mathematics
Problem
Five straight lines are given below. a. (i) Rewrite each equation (where possible) in slope–intercept form and state the slope of every line. For any line whose slope is not defined, mention that explicitly. b. (ii) List all distinct pairs of lines that are parallel. Justify your answer using the slopes found in part (i). c. (iii) List all distinct pairs of lines that are perpendicular. Justify your answer using the slopes found in part (i). d. (iv) The lines and intersect at the point . Determine the coordinates of , and write the equation of the line that passes through and is perpendicular to .Intuitive explanation
The key is recognizing that parallel lines have the same slope, while perpendicular lines have slopes whose product is . Start by rewriting each equation in slope-intercept form () to extract the slope clearly—this lets you quickly identify which lines are parallel to each other and which pairs are perpendicular. Vertical and horizontal lines are special cases: a vertical line has undefined slope and is perpendicular to any horizontal line.
Intuitive explanation
The key is recognizing that parallel lines have the same slope, while perpendicular lines have slopes whose product is . Start by rewriting each equation in slope-intercept form () to extract the slope clearly—this lets you quickly identify which lines are parallel to each other and which pairs are perpendicular. Vertical and horizontal lines are special cases: a vertical line has undefined slope and is perpendicular to any horizontal line.