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# lines - Lines and their equations The slope of a line...

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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.

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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 0 0 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 therefore the equation of the line.
Equation (ii) may also be written in the form: = - 2 y 6 - x 2, or = 2 y + x 4, or = y + x 2 2.

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