the line y=3x-1. find the equation of the line thats perpendicular to this line and pass through the point (4,-6) then the same but for perpendicular

Question

the line y=3x-1.Find the equation of the line that's perpendicular to this line and pass through the point (4,-6).Find the equation of the line that's parallel to the line and pass through the point (4,-6). 

Answer 1

There are two questions here:

1. Find line that's perpendicular to y=3x+1 which passes through (4,-6).

To find the equation of a perpendicular line, you have to find the slope and y-intercept of the line (the m and b in the equation y=mx+b). To solve for this, you first find the slope by following the rule for perpendicular lines. The slope of a line that's perpendicular to another line with a slope of m is -1/m (the negative reciprocal). So if your original line has an equation of y=3x+1, the slope of that line is 3. So the slope of a line perpendicular to that is -1/3.

Now we have the slope, but we need to find the full equation of the line that goes through (4,-6). To do that, we go back to the line equation y=mx+b. We have the slope now, so we know that y=(-1/3)x+b. The question actually provides you with a sample (x,y) coordinate pair that this line will contain. So we can plug those numbers into the line equation as well. The only variable left unsolved at that point is b.

-6 = (-1/3)(4) + b (multiply)

-6 = -4/3 + b (isolate b by adding 4/3 to both sides)

-6 + 4/3 = b (convert -6 to a fraction with a common denominator to 4/3)

-18/3 + 4/3 = b (combine terms)

b = -14/3

Combine the m and b values to form the equation of the perpendicular line using the same equation format of y=mx+b.

2. Find parallel line that passes through (4,-6)

This one is a bit easier than the previous one because of the definition of what a parallel line is: it's a line that will never intercept its parallel line(s). Thus, they always go in the same direction - or, in other words, they have the exact same slope. So you don't have to solve for the slope at all, you can use the same slope as the original equation. All you need to do then, is move to the "solve for b" portion of the previous example (as you have m=3, y=-6, x=4 already) to get the y-intercept value. Then, put the equation together using the m value you already know and the b value you calculated.

Good luck!

Answer 2

An important characteristic of a line is its slope. The slope of y = 3x + 1 is "3". When the equation is in the form y = mx + b then the slope is the number in front of the "x". (The "b" is the y-intercept or the location on the y-axis that the line crosses)

A line that is perpendicular to another line would have a slope that is the negative reciprocal (reciprocal means you flip the fraction over). So if a line has a slope of 2/3 then the line perpendicular to that has a slope of -3/2. If a line has the slope 3 (or 3/1) then the line perpendicular to it has the slope of -1/3.

A line that is parallel to another line has the same slope (but a different y-intercept). So a line that is parallel to a line with slope 3, also has a slope of 3.

Once you have the slope of a line, and you have a point on that line, you can "fill-in-the-blanks" of the point-slope form to get the equation. The point-slope form is:

y - y₁ = m(x - x₁) where "m" is the slope and (x₁, y₁) is the point. In this case, the point is (4, -6) so x₁ = 4 and y₁ = -6. The line perpendicular has m = -1/3 and the line parallel has m = 3. You just plug in the values.

Answer 3

For the line given, the slope is m = 3. A line perpendicular to that line has a slope of -1/m.

y+6 = -(x-4)/3

y = -x/3+4/3-6

y=-x/3-14/3