AMSCO'S Geometry. New York - Rye High School

298 Slopes and Equations of LinesSolutiona. The points A(2, 3), B(0, 1), and C(3, 7) lie on the same line if and only ifthe slope of AB is equal to the slope of BC.slope of AB 5 23 22 2 2 10slope of BC 5 1 0 2 2 735 24225 26235 25 2The slopes are equal. Therefore, the three points lie on the same line.b. Use the point-slope form of the equation of a line. Let (x, y) be any otherpoint on the line. You can use any of the points, A, B, or C, with (x, y) andthe slope of the line, to write an equation. We will use A(2, 3).y 2 (23)x 2 (22) 5 2y 1 3 5 2(x 1 2)y 1 3 5 2x 1 4y 5 2x 1 1Answers a. Since the slope of AB is equal to the slope of BC, A, B, and C lie on a line.b. y 2x 1AlternativeSolution(1) Write the slope-intercept form of anequation of a line:(2) Substitute the coordinates of A in thatequation:(3) Substitute the coordinates of C in thatequation:(4) Write the system of two equations from (2)and (3):y mx b3 m(2) b7 m(3) b3 2m b7 3m b(5) Solve the equation 3 2m b for b interms of m:(6) Substitute the value of b found in (5) for b inthe second equation and solve for m:(7) Substitute this value of m in either equationto find the value of b:The equation is y 2x 1.b 2m 37 3m b7 3m (2m 3)7 5m 310 5m2 mb 2m 3b 2(2) 3b 1