HFCC Math Lab Intermediate Algebra 1 SLOPE INTERCEPT AND POINT-SLOPE FORMS OF THE LINE

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1 HFCC Math Lab Intermediate Algebra SLOPE INTERCEPT AND POINT-SLOPE FORMS OF THE LINE THE EQUATION OF A LINE Goal I. Use the slope-intercept form of the line to write the equation of a non-vertical line A. given its -intercept and slope B. given its -intercept and another point on the line Goal II. Write the equation of a horizontal line through a given point. Goal III. Use the point-slope form of the line to write the equation of a non vertical line A. given its slope and a point on the line B. given two points on the line Goal IV. Write the equation of a vertical line through a given point. Goal V. Given the equation of a line, find the slope and the -intercept. TERMINOLOGY m = slope of the line b = the -intercept (where the line crosses the -axis) ( x, ), ( x, ) two given points on the line ( ma be the intercept points) I. SLOPE-INTERCEPT FORM OF A LINE: The equation of the line with slope m and -intercept b is given b = mx + b A.. Find the equation of the line with slope 5 and -intercept 7. m = 5, b = 7; = mx + b Therefore, = 5x + 7 is the equation of the line.. Find the equation of the line with slope m =, b = 0; = mx + b Therefore, and -intercept 0. x 0 x is the equation of the line. Revised /09

2 B.. Find the equation of the line with -intercept 6 and passing through the point (, ). First, we need to find the slope of the line, m: The line passes through the points (0, 6) and (, ). We can use the slope formula m where ( x, ) (0,6) and (x, ) (,) x x 6 m 0 Therefore, m = - and b = 6 = mx + b = -x Find the equation of the line with -intercept - and passing through the point (. -). First, we need to find the slope of the line, m: ( ) 0 m = 0 where ( x, ) (, ) &( x, ) (0, ) x x 0 Therefore, m = 0 and b = - = mx + b = 0x + - or = -. II. EQUATION OF HORIZONTAL LINE: The slope of a horizontal line is zero. The equation of a horizontal line through a point ( ( x, ) is.. Find the equation of the horizontal line passing through the point ( -, -5). = -5. Find the equation of the line having slope of 0 and -intercept of 9. Slope of 0 implies the line is horizontal. ( x, ) (0,9) so the equation is = 9.. Find the equation of the line passing through the points (, 5) and (-, 5). The slope of the line is = 0 where (, ) (,5) & (, ) (,5) m x x x x Therefore, the line is horizontal and has an equation = 5. Revised /09

3 III. POINT-SLOPE FORM OF A LINE: The equation of the line with slope m and passing through the point ( x, ) is given b the equation m( x x ). A.. Find the equation of the line with slope -6 and passing through the point (, 5). m 6; ( x, ) (,5) m( x x ) 5 6( x ) [You can rewrite the equation in slope-intercept form b solving for ] -5-6x 5 5 6x 5 6x 7.. Find the equation of the line with slope and passing through the point (-, -). m ; ( x, ) (, ) m( x x ) ( ) ( x ( )) or ( x ) [You can rewrite the equation in slope-intercept form b solving for ] x 6 x 6 x 5. B.. Find the equation of the line passing through the point (6, ) and (-, 7) First, we need to find the slope m: 7 m = - where ( x, ) (6,) & ( x, ) (,7) x x 6 8 m ; ( x, ) (6,) m( x x ) or ( x 6) - x 6 in slope-intercept form. Revised /09

4 . Find the equation of the line passing through the point (, 5) and (, -) First, we need to find the slope m: 5 9 m undefined where ( x, ) (,5) & ( x, ) (, ) x x 0 m is undefined, the line is vertical. Therefore, the equation of the line is x =. IV. EQUATION OF A VERTICAL LINE: The slope of a vertical line is undefined. The equation of a vertical line passing through the point ( x, ) is x x. Find the equation of the vertical line passing through the point (-7, 6). The equation of the line is x = -7.. Find the equation of the line with undefined slop and x-intercept 8. ( x, ) = (8, 0). Therefore, the equation of the line is x = 8.. Find the equation of the line passing through the points (, ) and (, -). First, we need to find the slope m: 5 m undefined where ( x, ) (,) &( x, ) (, ) x x 0 Therefore, the equation of the vertical line is x =. V. Given the equation of a non-vertical line, ou can find the slope and -intercept b rewriting the equation in slope-intercept form ( = mx+b) and then reading the values of m and b.. Find the slope and -intercept of the line x + = 5. First, we need to solve for : x 5 x x x 5 [subtract x from both sides] x 5 mx b, m = - and b = 5 Therefore, the slope is - and the -intercept (0, 5).. Find the slope and -intercept of the line x +6 =. First, we need to solve for : Revised /09

5 x 6 x x 6 x [subtract x from both sides] 6 x 6 x [divide both sides b 6] x mx b, m = - and b = Therefore, the slope is and the -intercept is (0, ). EXERCISES A. Write the equation of the line described in each of the exercises below. Write our answer in slope-intercept form if possible.. slope 5 and -intercept of -.. slope and -intercept.. - intercept of 7 and passing through the point (, ).. - intercept of 0 and passing through the point (, 5). 5. -intercept of - and x-intercept. 6. a horizontal line with -intercept slope and passing through the point (-5, ). 8. slope of - and passing through the point (, -). 9. slope of and x-intercept of a vertical line with x-intercept 5. Revised /09 5

6 . passing through the points (-, 5) and (6, -).. passing through (-, ) and (-6, 7).. passing through the points (-5, 0) and (6, 0).. x-intercept 5 and passing through the point (, ). 5. passing through the point (9, -) and parallel to the line with slope. 6. parallel to the -axis and passing through the point (-, ). 7. Passing through the point (, ) and perpendicular to the line whose slope is B. Find the slope and the -intercept of the given line. 8. 8x 9. x x. x.. x 5 ANSWERS AND SOLUTIONS.... x. 5 x. 7 x 7 since the slope m x 0 or 5 x since the slope m 5. 0 Revised /09 6

7 0 ( ) 5. x since the slope m since ( x 5) or x 7 (in slope-intercept form). 8. ( x ) or - x. 0 ( x 8) or x we used (x, ) (8, 0) x 5 since x = x ( x ) or x+ since m. 6 ( ) 9. ( x ) or x since the slope m 0, and passes through the point (-5, 0). 6 ( 5). First, we find the slope m: 0 m = = -6 where ( x, ) (5,0) and ( x, ) (, ) m( x x ) 0 6( x 5) 6x 0 5. ( x 9) or x ( m = ) N.B. If two lines have the same slope, ou can conclude that the two lines are parallel and vice versa. 6. Since the line is parallel to the -axis, then it is a vertical line. The equation is x = m, b = since 8x and b solving for, we obtain x. 8. Note here that two lines are perpendicular if the product of their slopes is -. Therefore if the slope of the given line is ½ then the slope of the line in question must be - since Revised /09 7

8 The line passes through the point (, ) with slope of - has equation ( x ) x x m, b. 0. m, b =- since x 8 and b solving for, we obtain x.. m, b =0 since x 0 which is = mx + b.. m 0, b =- since 0x (this is a horizontal line).. m is undefined (this is a vertical line) and no -intercept since the line is parallel to the -axis. *************************************************************** NOTE: You can get additional instruction and practice b going to the following websites: Man examples explaining in details how to write an equation of a line in slope-intercept form. This is a tutorial on how to find the slopes and equations of lines. A review of the main results concerning lines and slopes and then examples with detailed solutions are presented. Revised /09 8