Slope-Intercept Form of a Linear Equation Examples
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1 Slope-Intercept Form of a Linear Equation Examples. In the figure at the right, AB passes through points A(0, b) and B(x, y). Notice that b is the y-intercept of AB. Suppose you want to find an equation for AB. Let m represent the slope of the line. y b x 0 y b x mx = y b Use the definition of slope. AB passes through (x, y) and (0, b), so that x = x, x = 0, y = y, and y = b. Solve for y.. Slope-Intercept Form of a Linear Equation The slope-intercept form of the equation of a line is, where m is the slope and b is the y-intercept Let (x, y) and (x, y ) be points on y y line AB. Then the slope, m is. x x Therefore, m(x x ) = y y. Derivation of point-slope form. Suppose you are given the slope and y-intercept of a line. You can find an equation of the line by substituting these values into the slope-intercept form of the equation. For example, if the slope of a line is and its y-intercept is 4, an equation of the line is y = x + 4. The standard form of this equation is x + y = 4. Recall that the standard form of the equation of a line is Ax + By = C. If B 0, the slope and y-intercept can be found by changing the equation to slopeintercept form. Florida September 0, 00
2 4. Example Find the slope-intercept form and the standard form of the equation of the line that has a slope of and passes through (6, 7). First, substitute the slope and coordinates of the point into the slope-intercept form and solve for b. Substitute 7 for y, for m, and 6 for x. 7 = ( )(6) + b 7 = 4 + b = b Write the equation in slope-intercept form. y = x + The standard form of this equation is x y = -9 Substitute for m and for b. 5. Example find the slope-intercept form and the standard form of the equation of the line that has slope of and passes through the point (, ) First, substitute the slope and coordinates of the point into the slope-intercept form and solve for b. Substitute for y, for m, and for x. = ()( ) + b = + b 5 = b Write the equation in slope-intercept form. y = x + 5 The standard form of this equation is x y = 5 Substitute for m and 5 for b. Florida September 0, 00
3 6. Example Find the slope-intercept form and the standard form of the equation of the line having slope and passing through the point (, ). First, substitute the slope and coordinates of the point into the slope-intercept form and solve for b. Substitute for y, for m, and for x. = ( )( ) + b = + b Write the equation in slope-intercept form. Substitute for m 5 5 = b y = x + 5 and for b. The standard form of this equation is x + y = 5 7. Example Find the slope-intercept form of the equation of a line passing through (, 5) and (, 0) First use the two given points to find the slope of the line 5 0 Remember that m represents the slope of the line Next, substitute the slope of the coordinates of one point in the slope-intercept form and solve the equation for b. 5 = ( )( ) + b = b Using the point (, 0) would yield the same result. The slope-intercept form of the equation of the line is y = x + 8. Example Find the slope intercept form and the standard form of the equation of the line with x-intercept 4 and y-intercept. Find the slope for is (4, 0) and (0, ) Since y-intercept is x 6y = 4 Substitute into slopeintercept form 0 ( ) y =mx + b 4 0 y = x 6 6 Florida September 0, 00
4 9. Example Find the slope-intercept form and the standard form of the equation of the line passing through the points (, ) and (, ). Find the slope. ( ) ( ) Find y-intercept = () + b 5 = b substitute (, ) for x and y 5 y = x x y = 5 0. Example Find the slope and y-intercept of a line with equation in standard form. Recall that the standard form of the equation of a line is Ax + By = C. If B 0, the slope and y-intercept can be found by changing the equation to slopeintercept form. Ax + By = C By = Ax + C A C y = x + B B The slope is A C and the y-intercept is, for B 0 B B. Example Using the calculations from # 0 above, find the slope and y-intercept in the equation y = x + 4. The standard form of this equation is x + y = 4 A slope = B slope = = y-intercept = B C y-intercept = 4 = 4 If B = 0 in the standard form of the equation, the slope is undefined, the line has no y- intercept, and C 0. Florida September 0, 00
5 Name: Date: Class: Slope-Intercept Form of a Linear Equation Worksheet Find the slope-intercept form of the equation of the line for each slope, m, and each y- intercept, b.. 5, b =., b = , b = , b = 0., b = 7 State the slope and y-intercept of the graph of each equation 6. y = x y = x 8 9. y = x. y = 0.x + 6. y = x 8. y = x 4 0. y = 7x State the slope-intercept form of each equation.. x + 5y = 0 4. x y = x + 8y = x 0.4y = x y = 7 6. x = Find the slope-intercept form and the standard form of the equation of the line that satisfies the given conditions. 9. Slope =, passes through (6,4) 0. Slope = 4, passes through (, ). Slope = 5, passes through the origin. Passes through (6, ) and (8, 4). x-intercept =, y-intercept = 6 4. x-intercept = 0, y-intercept = Florida September 0, 00
6 Slope-Intercept Form of a Linear Equation Worksheet Key Find the slope-intercept form of the equation of the line for each slope, m, and each y-intercept, b.. 5, b = y = 5x., b = 4 y = x + 4., b = 7 y = x , b = 0 y =.5x 5. 0, b = 0 y = 0 State the slope and y-intercept of the graph of each equation 6. y = x + 5 slope =, y-intercept (0, 5) 7. y = x 8 slope =, y-intercept (0, 8) 8. y = x slope = 4, y-intercept (0, ) 4 9. y = x slope =, y-intercept (0, 0) 0. y = 7x y = 4x slope = 4, y-intercept (0, ). y = 0.x + 6 y = 0.x 6 slope = 0., y-intercept (0, 6). y = x y = x slope =, y-intercept (0, 0) State the slope-intercept form of each equation.. x + 5y = 0 5 x + 4. x y = 6 y = x 6 Florida September 0, 00
7 5. 4x + 8y = y = x x = no slope-intercept form x 0.4y = 0.4 y = x 8. 5 x y = 7 y = 5 x 7 Find the slope-intercept form and the standard form of the equation of the line that satisfies the given conditions. 9. Slope =, passes through (6,4) 4 = (6) + b = b y = x + x y = 0. Slope = 4, passes through (, ) = 4() + b = b y = 4x 4x y =. Slope = 5, passes through the origin 0 = 5(0) + b 0 = b y = 5x 5x y = 0. Passes through (6, ) and (8, 4) ( 4) 5 = (6) + b (6 8) 5 6 = b 5 y = x + 6 5x + y = Florida September 0, 00
8 . x-intercept =, y-intercept = = ( ) + b 6 = b y = x + 6 x y = 6 4. x-intercept = 0, y-intercept = slope undefined No slope-intercept form x = 0 Florida September 0, 00
9 Student Name: Date: Slope-Intercept Form of a Linear Equation Checklist. On questions thru 5, did the student find the slope-intercept form of each equation? a. Yes (5 points) b. 4 out of 5 (0 points) c. out of 5 (5 points) d. out of 5 (0 points) e. out of 5 (5 points). On questions 6 thru, did the student state the slope correctly? a. Yes (5 points) b. 6 our of 7 (0 points) c. 5 out of 7 (5 points) d. 4 out of 7 (0 points) e. out of 7 (5 points) f. out of 7 (0 points) g. out of 7 (5 points). On questions 6 thru, did the student state the y-intercept correctly? a. Yes (5 points) b. 6 our of 7 (0 points) c. 5 out of 7 (5 points) d. 4 out of 7 (0 points) e. out of 7 (5 points) f. out of 7 (0 points) g. out of 7 (5 points) 4. On questions thru 8, did the student state the slope-intercept form of each equation? a. Yes (5 points) b. 6 our of 7 (0 points) c. 5 out of 7 (5 points) d. 4 out of 7 (0 points) e. out of 7 (5 points) f. out of 7 (0 points) g. out of 7 (5 points) 5. On questions 9 thru 4, did the student find the slope-intercept form correctly? a. Yes (0 points) b. 5 out of 6 (5 points) c. 4 out of 6 (0 points) d. out of 6 (5 points) e. out of 6 (0 points) f. out of 6 (5 points) 6. On questions 9 thru 4, did the student find the standard form correctly? a. Yes (0 points) b. 5 out of 6 (5 points) c. 4 out of 6 (0 points) d. out of 6 (5 points) e. out of 6 (0 points) f. out of 6 (5 points) Total Number of Points Florida September 0, 00
10 Name: Date: Class: NOTE: The sole purpose of this checklist is to aide the teacher in identifying students that need remediation. It is suggested that teacher s devise their own point range for determining grades. In addition, some students need remediation in specific areas. The following checklist provides a means for the teacher to assess which areas need addressing.. Does the student need remediation in content (finding the slope-intercept form of an equation given the slope and y-intercept) for questions thru 5? Yes No. Does the student need remediation in content (identifying the slope and y-intercept in a linear equation) for questions 6 thru? Yes No. Does the student need remediation in content (rewriting linear equations into slope-intercept form) for questions thru 8? Yes No 4. Does the student need remediation in content (writing a linear equation in slopeintercept form given a point and the slope) for questions 9 thru 4? Yes No A B C D F 80 points and above 7 points and above 5 points and above points and above points and below Sample point range A B C 78 points and above 6 points and above points and above Florida September 0, 00
11 D F 4 points and above points and below Florida September 0, 00
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