MHB What Is the Equation of a Line Parallel to 4x + 5y = 20 Passing Through (0,0)?

[mathdad]

Find an equation of the line that is // to 4x + 5y = 20 and passes through the point (0,0). Write your answer in two forms: y = mx + b and Ax + By + C = 0.

The equation we want is parallel to the given equation. This means the slope is the same.

Steps:

1. Solve the given equation for y.

2. The slope is the coefficient of x in the equation found in step 1.

3. Plug the slope and the point (0,0) into the point-slope formula and solve for y.

4. Also write the equation of the line in the form
Ax + By + C = 0.

Correct?

 

[MarkFL]

Suppose we generalize the problem to finding a line parallel to $Ax+By=C$ which passes through $\left(x_1,y_1\right)$

Note: $$B\ne0$$

We can readily see that all lines of the form $Ax+By=k$ where $k\in\mathbb{R}$ will be parallel to the given line. So, in order to determine $k$, we can write:

$$Ax_1+By_1=k$$

Thus, our line is:

$$Ax+By=Ax_1+By_1$$

or:

$$Ax+By+\left(-Ax_1-By_1\right)=0$$

And in slope-intercept form, this is:

$$y=-\frac{A}{B}x+\frac{Ax_1+By_1}{B}$$

Now we have formulas to use for this kind of problem. (Yes)

 

[mathdad]

Suppose we generalize the problem to finding a line parallel to $Ax+By=C$ which passes through $\left(x_1,y_1\right)$

Note: $$B\ne0$$

We can readily see that all lines of the form $Ax+By=k$ where $k\in\mathbb{R}$ will be parallel to the given line. So, in order to determine $k$, we can write:

$$Ax_1+By_1=k$$

Thus, our line is:

$$Ax+By=Ax_1+By_1$$

or:

$$Ax+By+\left(-Ax_1-By_1\right)=0$$

And in slope-intercept form, this is:

$$y=-\frac{A}{B}x+\frac{Ax_1+By_1}{B}$$

Now we have formulas to use for this kind of problem. (Yes)

Cool but are my steps incorrect?

 

[MarkFL]

Cool but are my steps incorrect?

Let's use the generalized problem, and follow your steps...

1. Solve the given equation for y.

I would state the first step as:

1.) Express the given equation in slope-intercept form.

$$y=-\frac{A}{B}x+\frac{C}{B}$$

2. The slope is the coefficient of x in the equation found in step 1.

We then identify the slope as:

$$m=-\frac{A}{B}$$

3. Plug the slope and the point $\left(x_1,y_1\right)$ into the point-slope formula and solve for y.

Again, instead of "solve for $y$" I would use "express in slope-intercept form."

$$y-y_1=-\frac{A}{B}\left(x-x_1\right)$$

$$y=-\frac{A}{B}x+\frac{Ax_1+By_1}{B}\quad\checkmark$$

4. Also write the equation of the line in the form
Ax + By + C = 0.

$$y=-\frac{A}{B}x+\frac{Ax_1+By_1}{B}$$

$$By=-Ax+Ax_1+By_1$$

$$Ax+By+\left(-Ax_1-By_1\right)=0\quad\checkmark$$

As you can see, we have the same formulas.

 

[mathdad]

What is A, B, x_1 and y_1 in your method?

 

[MarkFL]

What is A, B, x_1 and y_1 in your method?

They come from the generalized problem:

Suppose we generalize the problem to finding a line parallel to $Ax+By=C$ which passes through $\left(x_1,y_1\right)$

Note: $$B\ne0$$

 

[mathdad]

They come from the generalized problem:

I am preparing a reply to your answers in our PM discussion. I will work on this question when time allows. I am off tomorrow and Tuesday. A little break from work.

 

[mathdad]

Some of you may ask: WHAT DOES RTCNTC do on his days off? If you said MATH, you are right. Why not math? I have no girlfriend, my family is scattered, no one wants to date a fat, bald headed, 52 year old middle-aged man. So, math is good for me. lol

 

[MarkFL]

Some of you may ask: WHAT DOES RTCNTC do on his days off?

My main question regarding RTCNTC is:

When is he going to start using $\LaTeX$?

 

[mathdad]

I will do my best to start using LaTex.

 

[mathdad]

My main question regarding RTCNTC is:

When is he going to start using $\LaTeX$?

Can you show me how to solve this problem using the generalized formulas? I am confused about what to plug into the formulas.

 

[MarkFL]

Can you show me how to solve this problem using the generalized formulas? I am confused about what to plug into the formulas.

Okay, we have derived the formulas:

$$y=-\frac{A}{B}x+\frac{Ax_1+By_1}{B}\tag{1}$$

$$Ax+By+\left(-Ax_1-By_1\right)=0\tag{2}$$

We are given the line:

$$4x+5y=20$$

and the point:

$$(0,0)$$

And so we identify:

$$A=4,\,B=5,\,C=20,\,x_1=0,\,y_0=0$$

Plugging these values into (1), we obtain:

$$y=-\frac{4}{5}x+0$$

And into (2), we obtain:

$$4x+5y+0=0$$

 

[mathdad]

Okay, we have derived the formulas:

$$y=-\frac{A}{B}x+\frac{Ax_1+By_1}{B}\tag{1}$$

$$Ax+By+\left(-Ax_1-By_1\right)=0\tag{2}$$

We are given the line:

$$4x+5y=20$$

and the point:

$$(0,0)$$

And so we identify:

$$A=4,\,B=5,\,C=20,\,x_1=0,\,y_0=0$$

Plugging these values into (1), we obtain:

$$y=-\frac{4}{5}x+0$$

And into (2), we obtain:

$$4x+5y+0=0$$

Wow! That was quick! I tried using the QUICK LATEX box but it is not working on my phone. Where is the LaTex lesson page for the MHB?

 

[MarkFL]

Wow! That was quick! I tried using the QUICK LATEX box but it is not working on my phone. Where is the LaTex lesson page for the MHB?

Are you sure the cursor is in the post editor when you click the commands/symbols in the Quick $\LaTeX$?

 

[mathdad]

I will post a question using the Quick LaTex just for practice.

 

[MarkFL]

Are you sure the cursor is in the post editor when you click the commands/symbols in the Quick $\LaTeX$?

Still wondering about this...(Wondering)

 

[mathdad]

Still wondering about this...(Wondering)

I click the QUICK LATEX symbol that I need to use. It appears in the reply box but not when posting my reply.

 

[MarkFL]

I click the QUICK LATEX symbol that I need to use. It appears in the reply box but not when posting my reply.

You click a symbol/command, and it appears in the editor, but when you post your reply, it disappears?

 

[mathdad]

You click a symbol/command, and it appears in the editor, but when you post your reply, it disappears?

Yes.

 

[MarkFL]

Yes.

Are you sure you're talking about the editor, and not the $\LaTeX$ Live Preview?

 

[mathdad]

I click on the Commands for what I need. The symbol appears in the reply box but after clicking submit, the LaTex form does not display.

 

[MarkFL]

I click on the Commands for what I need. The symbol appears in the reply box but after clicking submit, the LaTex form does not display.

Are you wrapping the code in $$$$ tags?

 

[mathdad]

Are you wrapping the code in $$$$ tags?

No. Can you explain how this is done?

 

[MarkFL]

No. Can you explain how this is done?

The easiest way is to first click the $$\sum$$ button on the editor toolbar, and once you click that the $$$$ tags will be added to your post content at the current cursor location, and the cursor will be in between the tags, ready for you to add your code. :)

 

[mathdad]

The easiest way is to first click the $$\sum$$ button on the editor toolbar, and once you click that the $$$$ tags will be added to your post content at the current cursor location, and the cursor will be in between the tags, ready for you to add your code. :)

Let me try again.

- - - Updated - - -

Let me try e^x.

$$e^{x}$$

- - - Updated - - -

It works!

Let me try again.

I will try e^(x + 1).

$$e^{x + 1}$$

 

[MarkFL]

 

[mathdad]

Moving on.....

 

[MarkFL]

Moving on.....

http://cdn2.bigcommerce.com/server4900/364bb/products/127384/images/91828/264870__90461.1342529542.380.500.jpg?c=2

 

[mathdad]

I will post two daily questions only. In so doing, I can spend more time per question. As you know, some of these David Cohen questions are very involved.