Answer:
1) Parallel line: y = 4x + 1
4) Perpendicular line: y = -5x + 9
6) Perpendicular line: y = 3x + 2
Step-by-step explanation:
Note:
I will only provide solutions to question 1, 4, and 6, in accordance with Brainly's guidelines. Since the given problem essentially involves the same type of questions, the solutions that I will provide should allow you to apply the same techniques in solving for the remainder of the questions.
Definitions:
Parallel lines have the same slope.
Perpendicular lines have negative reciprocal slopes. This means that when you multiply the slopes of two lines that are perpendicular from each other will result in a product of - 1.
In other words, if the slope of a line is m₁, and the slope of the other line is m₂, then it means that they are perpendicular from each other if: m₁ × m₂ = - 1
Solutions:
1) y-intercept = 1, parallel to y = 4x - 3.
According to the definitions provided on parallel lines, since the given linear equation has a slope, m = 4, then it means that the other line parallel to it must have the same slope.
Given the y-intercept of the other line as b = 1, and its slope, m = 4, then the equation of the line parallel to y = 4x + 3 is:
Equation of Parallel line: y = 4x + 1.
4) y-intercept = 9, perpendicular to y = ⅕x + 4.
Given the linear equation, y = ⅕x + 4, where its slope, m₁ = ⅕
Then it means that the negative reciprocal of ⅕ must be ⇒ [tex]\large\mathsf{-\frac{5}{1}\:\:or\:\:-5}[/tex]
Because: m₁ × m₂ = - 1
⅕ × -5 = -1
Therefore, if the y-intercept of the other line is b = 9, and its slope is m₂ = -5, then the equation of the line perpendicular to y = ⅕x + 4 is:
Perpendicular line: y = -5x + 9.
6) Through point (2, 8) and perpendicular to y = - ⅓x - 8
Given the equation of the first line, y = -⅓x - 8, where its slope, m₁ = -⅓.
Then it means that the slope of the other line must be m₂ = [tex]\large\mathsf{\frac{3}{1}\:\:or\:\:3}[/tex] because:
m₁ × m₂ = - 1
-⅓ × 3 = -1
Next, substitute the slope of the other line, m₂ = 3, and the values of the given point, (2, 8) into the slope-intercept form, y = mx + b, in order to solve for the y-intercept of the other line.
y = mx + b
8 = 3(2) + b
8 = 6 + b
Subtract 6 from both sides to isolate b:
8 - 6 = 6 - 6 + b
2 = b
Therefore, the equation of the line perpendicular to y = - ⅓x - 8 is:
Perpendicular line: y = 3x + 2.
