a. Slope is 1/3.
b. Point-slope form: y - 1 = (1/3)(x - 1).
c. Slope-intercept form: y = (1/3)x + 2/3.
d. Parallel line: y + 1 = (1/3)(x - 3).
e. Perpendicular line: y - 7 = -3(x + 2).
a. Slope of the Line:
The slope (m) is calculated using the formula: m = (change in y) / (change in x). The slope is (2 - 1) / (4 - 1) = 1/3.
b. Equation in Point-Slope Form:
Using the point-slope form y - y1 = m(x - x1) with the given point (1, 1):
y - 1 = (1/3)(x - 1).
c. Equation in Slope-Intercept Form:
Simplifying the point-slope form equation:
y = (1/3)x + 2/3.
d. Equation of Line Parallel to Given Line:
Since parallel lines have the same slope, the equation will have slope 1/3 and pass through (3, -1):
y - (-1) = (1/3)(x - 3).
e. Equation of Line Perpendicular to Given Line:
Perpendicular lines have negative reciprocal slopes. The given line's slope is 1/3, so the perpendicular slope is -3.
Using the point-slope form with (-2, 7):
y - 7 = -3(x + 2).