Answer:
a. The equation of the parallel line to the given line is y = -4x + 19
b. The equation of the perpendicular line to the given line is y = [tex]\frac{1}{4}[/tex] x + 2
Step-by-step explanation:
Parallel lines have the same slopes
- If the slope of one of them is m, then the slope of the other is m
The product of the slopes of the perpendicular lines is -1
- If the slope of one of them is m, then the slope of the other is [tex]-\frac{1}{m}[/tex]
- To find the slope of a perpendicular line to a given line reciprocal the slope of the given line and change its sign
The rule of the slope is m = [tex]\frac{y2-y1}{x2-x1}[/tex] , where
- (x1, y1) and (x2, y2) are the points on the line
The form of the equation of a line is y = m x + b, where
Let us solve the question
∵ The given line passes through points (1, 6) and (2, 2)
∴ x1 = 1 and y1 = 6
∴ x2 = 2 and y2 = 2
→ Substitute them in the rule of the slope to find it
∵ m = [tex]\frac{2-6}{2-1}=\frac{-4}{1}=-4[/tex]
∴ The slope of the given line is -4
a.
∵ The line is parallel to the given line
∴ Their slopes are equal
∵ The slope of the given line = -4
∴ The slope of the parallel line = -4
→ Substitute its value in the form of the equation above
∴ y = -4x + b
→ To find b substitute x and y in the equation by the coordinates
of any point on the line
∵ The parallel line passes through the point (4, 3)
∴ x = 4 and y = 3
∵ 3 = -4(4) + b
∴ 3 = -16 + b
→ Add 16 to both sides
∴ 3 + 16 = -16 + 16 + b
∴ 19 = b
→ Substitute it in the equation
∴ y = -4x + 19
The equation of the parallel line to the given line is y = -4x + 19
b.
∵ The line is perpendicular to the given line
∴ The product of their slopes is -1
→ Reciprocal the slope of the given line and change its sign
∵ The slope of the given line = -4
∴ The slope of the perpendicular line = [tex]\frac{1}{4}[/tex]
→ Substitute its value in the form of the equation above
∴ y = [tex]\frac{1}{4}[/tex] x + b
→ To find b substitute x and y in the equation by the coordinates
of any point on the line
∵ The perpendicular line passes through the point (4, 3)
∴ x = 4 and y = 3
∵ 3 = [tex]\frac{1}{4}[/tex] (4) + b
∴ 3 = 1 + b
→ Subtract 1 from both sides
∴ 3 - 1 = 1 - 1 + b
∴ 2 = b
→ Substitute it in the equation
∴ y = [tex]\frac{1}{4}[/tex] x + 2
The equation of the perpendicular line to the given line is y = [tex]\frac{1}{4}[/tex] x + 2
