+60 Write the equations of the lines, representing the following conditions, in the form y = mx + b, where m is the slope and b is the y-intercept: Part A: Passes through (4, 3) and m = start fraction two over five end fraction Part B: Passes through (−4, 2) and (2, −5) Part C: m = start fraction three over five end fraction and y-intercept = −4 Part D: x-Intercept = 7 and y-intercept = −2 Part E: Passes through (−2, 2) and parallel to 4x − 3y − 7 = 0
+23245 Slope-intercept form of a straight line: y = mx + b
Point-slope form of a straight line: y - y1 = m(x - x1)
A) Through (4,3 ) and m = 2/5
---> Use the point-slope form with m = 2/5, x1 = 4, and y1 = 3: y - 3 = (2/5)(x - 4)
---> Cross-multiply: 5(y - 3) = 2(x - 4) ---> 5y - 15 = 2x - 8 ---> 5y = 2x + 7 ---> y = (2/5)x + (7/5)
B) Through (-4,2) and (2,-5)
---> First: find the slope: m = (-5 - 2) / (2 - -4) = -7/6
---> Use the point-slope form (using either point): y - 2 = (-7/6)(x - -4)
---> 6(y - 2) = -7(x + 4) ---> 6y - 12 = -7x - 28 ---> 6y = -7x - 16 ---> y = (-7/6)x - 8/ 3
C) m = 3/5 and y-intercept -4:
---> Use the slope-intercept form: y = (3/5)x - 4
D) x-int = 7 and y-int = -2:
---> x-int = 7 ---> (7,0)
y-int = -2 ---> (0,-2)
---> First: find the slope: m = (-2 - 0) / (0 - 7) = -2/-7 = 2/7
---> Use the slope-intercept form: y = (2/7)x - 2
E) Passes through (-2,2) and is parallel to 4x - 3y -7 = 0:
---> All parallel lines have the same x-coefficient and the same y-coefficient; so, all lines parallel to 4x - 3y -7 = 0 have the form: 4x - 3y + k = 0 (for some value of k).
---> To find the value of k replace x with -2 and y with 2:
---> 4(-2) - 3(2) + k = 0 ---> -8 - 6 + k = 0 ---> -14 + k = 0 ---> k = 14
---> Equation: 4x - 3y - 14 = 0
