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Unformatted text preview: Line s and their equations The slope of a line . Consider the line L that passes through the points ( ) P , 1 1 and ( ) Q , 2 4 . In moving from P to Q along the line L, the horizontal distanced travelled is 1 unit and the vertical distance travelled is 3 units. In fact whenever one moves along the line so that the horizontal distance traversed is 1 unit, the line rises through a distance of 3 units. For this reason we say that the slope of this line is 3. In general, if a line goes upwards in the direction from left to right, its slope is the distance that the line rises in moving to the right (parallel to the x axis ) through a distance of 1 unit. If a line goes down from left to right it has the negative slope given by taking the negative of the distance that the line falls when following the line through a horizontal distance of 1 unit. The line L considered previously passes through the point ( ) R , 3 7 , and the slope can also be obtained from = TR PT 6 2 = 3. In general, if a line goes upwards from left to right, its slope can be obtained as: vertical distance traversed horizontal distance traversed . This is often given briefly as = slope rise run . If a line goes downwards from left to right so that the line falls when moving horizontally, then the vertical distance is given a minus sign so that the slope will be negative. The slope formula . Suppose that we are given two distinct points A ( ) , x 1 y 1 and B ( ) , x 2 y 2 in the coordinate plane where the second point B is above and to the right of the first point A. Then the line passing through A and B moves upwards through a vertical distance - y 2 y 1 while travelling horizontally through a distance - x 2 x 1 . The slope of the line is given by - y 2 y 1- x 2 x 1 . ____ It turns out that this formula always gives the slope of the line through A and B no matter where the two points are in relation to each other, as long as they are not in the same vertical line. For, example, if B is to the right of and below A, then the line falls through a distance - y 1 y 2 while moving horizontally through a distance - x 2 x 1 , so its slope is = - - y 1 y 2- x 2 x 1- y 2 y 1- x 2 x 1 . The point-slope equation of a line . Example 1 : Find the equation of the line that passes through the point ( ) , 2 3 and has slope 1 2 . Solution : Suppose that ( ) P , x y is a point on the line which is different from ( ) A , 2 3 . Then the slope of AP is - y 3- x 2 . Since the slope of the line is 1 2 , we have: = - y 3- x 2 1 2------- (i), that is, = - y 3 1 2 ( )- x 2 ------- (ii). Note that the coordinates of the point ( ) A , 2 3 do not satisfy the equation (i) because the left hand side of (i) is the meaningless expression when = x 3 and = y 2. However, they do satisfy equation (ii) because the left and right sides of (ii) are both equal to 0 when = x 2 and = y 3. Hence (ii) is an equation satisfied by the coordinates of all points on the line, (and by the coordinates of no other points). It is 3....
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