Linear Equation thru point (-2,-4), perpendicular to the line y = -3x-5.

[sean1]

I am very confused about this problem that was on my quiz:

Find the equation of the line that passes through the point (-2,-4) and is perpendicular to the line -3x-5.

I know that line 1 has a y intercept of (0,-5) and it's slope is -3/1. I do not know how I would use this information to determine what the equation is of the other line

 

[JeffM]

I am very confused about this problem that was on my quiz:

Find the equation of the line that passes through the point (-2,-4) and is perpendicular to the line -3x-5.

I know that line 1 has a y intercept of (0,-5) and it's slope is -3/1. I do not know how I would use this information to determine what the equation is of the other line

You need to know the fact that if the slope of a given line is a then the slope of any line perpendicular to that given line is - (1/a).

So what next?

 

[sean1]

You need to know the fact that if the slope of a given line is a then the slope of any line perpendicular to that given line is - (1/a).

So what next?

So the equation so far is 1/3x+?

I am drawing a blank on how to find the y intercept of line 2 with this information

 

[sean1]

So the equation so far is 1/3x+?

I am drawing a blank on how to find the y intercept of line 2 with this information

This is what I came up with but it does not seem correct to me:

y=1/3x+b
-4=1/3(-2)+b
-4=-2/3=b
b=-10/3

meaning line 2: 1/3x+-10/3

 

[JeffM]

This is what I came up with but it does not seem correct to me:

y=1/3x+b
-4=1/3(-2)+b
-4=-2/3=b
b=-10/3

meaning line 2: 1/3x+-10/3

Your answer is 99.5% correct. Why do you doubt yourself? Technically, you should not stick operator symbols together as you did, but it is obvious what you meant. I'd give you full credit for that answer if I were grading it.

\(\displaystyle y = (1/3) x - 10/3.\) That is correct.

As a matter of style, I would express the same mathematical fact as

\(\displaystyle y = \dfrac{x - 10}{3}\). Very easy to work with.

But your teacher may well prefer your form because it clearly shows the slope of the perpendicular as the additive inverse of the multiplicative inverse of the slope of the original line.

 

[lookagain]

As a matter of style, I would express the same mathematical fact as

\(\displaystyle y = \dfrac{x - 10}{3}\). Very easy to work with.

"As a matter of style," the student should not even consider writing the equation of the line for the instructor like that,
because it doesn't fall under standard styles of different types of equations of lines covered in algebra regarding
equations of lines passing trough two points, passing through a point and parallel to a give line, passing through a
point and perpendicular to a given line, or the like.

It would be a different case if the instructor wasn't discussing lines, and had the equation \(\displaystyle x - 3y = - 10\) and asked
to solve it for y. Then, the equation given in the quote box would be one of the expected answers in that context.

 

[mmm4444bot]

Another way is to use the Point-Slope form. We call this form "Point-Slope" because we use it when we know a point and the slope.

y - y₁ = m(x - x₁)

You know the slope, and you know the coordinates of one point.

m = 1/3

x₁ = -2

y₁ = -4

Substitute the known values:

y + 4 = 1/3(x + 2)

That ought to be an acceptable answer because the exercise does not specify that you need to report the equation using any particular form. :cool:

 

[sean1]

Your answer is 99.5% correct. Why do you doubt yourself? Technically, you should not stick operator symbols together as you did, but it is obvious what you meant. I'd give you full credit for that answer if I were grading it.

\(\displaystyle y = (1/3) x - 10/3.\) That is correct.

As a matter of style, I would express the same mathematical fact as

\(\displaystyle y = \dfrac{x - 10}{3}\). Very easy to work with.

But your teacher may well prefer your form because it clearly shows the slope of the perpendicular as the additive inverse of the multiplicative inverse of the slope of the original line.

Thanks! I think I doubted myself so much because the exercises in the text book worked out to whole numbers but it looks like my professor slapped this together for the quiz without giving me the same courtesy. Thanks again for walking me through this one :cool:

 

[sean1]

Another way is to use the Point-Slope form. We call this form "Point-Slope" because we use it when we know a point and the slope.

y - y₁ = m(x - x₁)

You know the slope, and you know the coordinates of one point.

m = 1/3

x₁ = -2

y₁ = -4

Substitute the known values:

y + 4 = 1/3(x + 2)

That ought to be an acceptable answer because the exercise does not specify that you need to report the equation using any particular form. :cool:

Thanks for breaking that down. That is actually really easy for me to understand. It is a much easier way to write that, I will double check to make sure I can express it that way on my upcoming final. Thanks again :cool: