EQUATION FORMS OF A LINE Machinery's Handbook, 31st Edition
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y 2 y 1 – x 2 x 1 – = -------- into the point-slope form and rearranging
Point-Point Form: Substituting m
terms gives y y ) , the point-point form . Slope-Intercept Form: Another often-used form is y = mx + b , where m is the slope and b is the y -intercept. (That is, when x = 0, then y = m (0) + b = b .) If the pre- vious example is rearranged and solved for y , then the slope-intercept form is arrived at: y x x x =− −+=− + =− + + 2 3 x x y y − 1 2 1 2 ( x x − = 1 − − 1 2 3 . Standard Form of a Line: In vector representations (see Complex Numbers on page 59), lines usually take the standard form, Ax + By = C , where A, B, C . For example, 7 x – 3 y = 4 is a line in standard form, with A = 7, B = –3, and C = 4. Example 4: What is the standard form equation of a line AB between points A (4, 5) and B (7, 8)? Solution: Using the point-point form of the line, where (4, 5) is ( x 1 , y 1 ) and (7, 8) is ( x 2 , y 2 ), y – y 1 = ( x – x 1 ) 8 3 2 3 11 3 4 1 ) 1 (
y 1 – y 2 x 1 – x 2
( x – 4) 5 – 8 4 – 7
y – 5 =
y – 5 = x – 4 x – y = –1 Example 5: Find the slope-intercept equation of a line passing through the points (3, 2) and (5, 6). The y -intercept is the intersection point of the line with the y -axis. Solution: First, find the slope: m ∆ y ∆ x --- 6– 2 5– 3 ------ 4 2 -- 2 = = = = The slope-intercept form of the line is y = 2 x + b , and the value of the constant b can be determined by substituting the coordinates of a point on the line into the general form. Using the coordinates of the point (3, 2) gives 2 = (2)(3) + b and rearranging, b = 2 - 6 = - 4. As a check, using another point on the line, (5, 6), yields the same result, y = 6 = (2)(5) + b and b = 6 - 10 = - 4. The equation of the line, therefore, is y = 2 x - 4, indicating that line y = 2 x - 4 intersects the y -axis at point (0, - 4), the y -intercept. Example 6: Use the point-slope form to find the equation of the line passing through the point (3, 2) and having a slope of 2. y – 2 2 x – 3 ( ) = y 2 x = – 6+2 y 2 x = – 4 The slope of this line is positive and crosses the y -axis at the y -intercept, point (0, - 4). Parallel Lines: The two lines, l 1 and l 2 , are parallel if they have the same slope, that is, if m 1 = m 2 .
Fig. 5. (a) Parallel lines, l 1 and l 2 .
(b) Perpendicular lines, l 1 and l 2 .
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