Adding 'x' to 'xy' for Resultant <=1: A Table Guide

[pairofstrings]

TL;DR Summary

How do I add x + xy =1?

Hello.

How do I add 'x' to 'xy', and the resultant should be <= 1?
x + xy = 1

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PEMDAS:
Step 1: Multiply 'x' and 'y' (2^nd term in the equation)
Step 2: Add 1^st term to 2^nd term of the equation

Step 1 plus Step 2 should yield value <=1.
I know that the curve look like this:

'xy' will give me a curve to which I need to add 'x' and this graph is the output after I add 'x' to equation:
x + xy = 1.

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How do I get the resultant without simplifying like the following:
x + xy = 1
x (1 + y) = 1 ;
1 + y = 1/x
Then I could plug in numbers into 'x' to get y..
Table:
x | y
1 0

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Thanks.

 

[Drakkith]

How do I get the resultant without simplifying like the following:
x + xy = 1
x (1 + y) = 1 ;
1 + y = 1/x
Then I could plug in numbers into 'x' to get y..
Table:
x | y
1 0

I'm not quite sure what you're asking, as that's exactly how to solve this.
Are you just looking for another method, or have I misunderstood you?

 

[symbolipoint]

I'm not sure what you hope for here, but infinite possible x and y combinations. Try solve for y in terms of x. If pick x=1/2, then y=1. Now try a check.
x+xy=1
1/2+(1/2)*1=1

And you can easily find other combinations.

 

[pairofstrings]

Here's a little more detail:
The equation to understand is: x + xy = 1.
xy = 1 graph is known but how do I make an impact on xy = 1 by doing addition of 'x' to xy = 1?

How do I add 'x' to 'xy = 1' so that I get the curve below - the equation of the curve below is: x + xy = 1

The graph for the equation xy = 1 is:

I add 'x' to 'xy' to do some changes to the curve xy = 1.
The changes can be seen in first picture: x + xy = 1.
Second picture: xy = 1 -- no applied changes.

Thanks.

 

[Gaussian97]

If I understand correctly, you can think it like this:
If you know what the graph of $xy=1$ looks like, then notice that by changing $y \to y+1$ you obtain
$$xy=1 \to x(y+1)=1 \to x+xy=1$$
Then you only need to be familiar with the fact that doing a substitution like $y \to y+1$ affects your graph by "translating 1 unit down".
You can now look at the graphs you have drawn, and convince yourself that indeed they are identical, but one is 1 unit lower than the other.

This is an easy way to obtain the graph for $x+xy=1$ starting from $xy=1$, I don't know if that's what you were asking about.

 

[pbuk]

You have (almost) worked out for yourself that you can rewrite $x + yx = 1$ as $y = \frac 1 x - 1$. It should therefore not be any surprise that the graph of $x + yx = 1$ is the same as the graph of $y = \frac 1 x$ shifted down by 1.

You have made a number of posts here with curves plotted on Desmos or similar together with questions that show that you don't really understand how the algebraic expression of a function and its graph relate, and don't actually make sense and are therefore impossible to answer.

Have you studied curve sketching at school? If not, there are plenty of tutorials you can search for. This will help you understand much better than playing around with tools like Desmos.

Also, in your original question you ask that "the resultant should be less than or equal to 1" but I don't see where "less than or equal" fits in with the rest of your question. Do you understand how inequalities like this relate to parts of a graph?

 

[pairofstrings]

If I understand correctly, you can think it like this:
If you know what the graph of xy=1 looks like, then notice that by changing y→y+1 you obtain
xy=1→x(y+1)=1→x+xy=1
Then you only need to be familiar with the fact that doing a substitution like y→y+1 affects your graph by "translating 1 unit down".

This is little bit how I want to understand other complicated curves.

Also, in your original question you ask that "the resultant should be less than or equal to 1" but I don't see where "less than or equal" fits in with the rest of your question.

The points in the curve are -1 ≤ x ≤ 1, and -1 ≤ y ≤ 1.
The points should not go beyond -1 ≤ x ≤ 1, and -1 ≤ y ≤ 1.

Similar to the equation of a circle: x^2 + y^2 = 1.

You have (almost) worked out for yourself that you can rewrite x+yx=1 as y=1x−1. It should therefore not be any surprise that the graph of x+yx=1 is the same as the graph of y=1x shifted down by 1.

Is it necessary to rewrite the equation?
Is there any other way to write or understand equation?
Is this how I can build complex equations?

How I am trying to understand equation: x + xy =1
My post #4 has a curve that says how 'x' or 'y' increases or decreases as I plot xy = 1.

If 'x' is increasing in the 2^nd term of the equation then 'y' should decrease. My question is how, and when 'x' in the 1^st term of the equation going to contribute to the change? What is the first term doing to xy = 1 and how it is doing it?

Thanks.

 

[pbuk]

This is little bit how I want to understand other complicated curves.

As I said above, this is not the way to gain understanding. It's like trying to learn addition by saying "I found out that 2 + 2 = 4 by entering it into a calculator but what numbers do I have to add to get 6?". Search for "curve sketching" to learn the basics.

The points in the curve are -1 ≤ x ≤ 1, and -1 ≤ y ≤ 1.
The points should not go beyond -1 ≤ x ≤ 1, and -1 ≤ y ≤ 1.

Then you need to specify that you only want to draw points in that range in whatever software you are using (but I'm not sure Desmos has that option).

Similar to the equation of a circle: x^2 + y^2 = 1.

No, it is NOT similar to that equation. All values of x and y that satisfy that equation lie in the range [-1, 1] whereas in x + xy = 1 the values have no upper or lower bounds.

Is it necessary to rewrite the equation?

In order to understand what its graph looks like, yes. This is the first thing that you will learn when you study curve sketching: rewrite the equation in the form y = f(x) (or x = f(y)) if it is possible to do so.

Is there any other way to write or understand equation?

There is no other way to understand anything except by learning about it as I have explained above.

How I am trying to understand equation: x + xy =1
My post #4 has a curve that says how 'x' or 'y' increases or decreases as I plot xy = 1.
If 'x' is increasing in the 2^nd term of the equation then 'y' should decrease. My question is how, and when 'x' in the 1^st term of the equation going to contribute to the change? What is the first term doing to xy = 1 and how it is doing it?

But this isn't helping you understand is it? That is because when you have x and y terms on the same side of the equation it is very difficult to understand what is going on.

 

[Gaussian97]

Then you need to specify that you only want to draw points in that range in whatever software you are using (but I'm not sure Desmos has that option).

I know it's not really the main topic, but just in case the OP is interested, it is possible to restrict the domain and range of the graphs with Desmos using curly brackets: https://www.desmos.com/calculator/yywyn6pwba

 

[pairofstrings]

But this isn't helping you understand is it? That is because when you have x and y terms on the same side of the equation it is very difficult to understand what is going on.

Is this how I can build complex equations?

So, if I have to build complex equation that is in 'x' and 'y' then I need to gather data points that has both 'x' and 'y'. Then build equation? How? I need to try to fit the values into some curve? How? Interpolation?

Thanks.

 

[Mark44]

Summary:: How do I add x + xy =1?

PEMDAS:
Step 1: Multiply 'x' and 'y' (2nd term in the equation)
Step 2: Add 1st term to 2nd term of the equation

PEMDAS has absolutely zero to do with this.

The equation to understand is: x + xy = 1.
xy = 1 graph is known but how do I make an impact on xy = 1 by doing addition of 'x' to xy = 1?

This is the wrong question to ask. The equation x + xy = 1 has two variables, x and y, so one way to find solutions is to solve for one of the variables (as @pbuk did), and substitute values into the other variable. Your equation is equivalent to y = 1/x - 1, so you can find points on the graph by putting in values for x, and calculating the resulting y values.

You have made a number of posts here with curves plotted on Desmos or similar together with questions that show that you don't really understand how the algebraic expression of a function and its graph relate, and don't actually make sense and are therefore impossible to answer.

Have you studied curve sketching at school?

In one of the other threads by the OP, I recommended studying or reviewing some basic curves: straight lines, quadratics, and a few other graph types. These concepts are usually presented in a precalculus course.

 

[pairofstrings]

PEMDAS has absolutely zero to do with this.

Then how do I know if x + xy is really equal to one?

 

[Mark44]

The thread title is "Add x + xy = 1" .

There really are two parts here:

1. Add x + xy
   This is straightforward provided that you have a value for x and a value for y. Since there is no equation, you can pick any value for either variable. The only thing to keep in mind is that you have to multiply x and y first, and then add the value of x. This is due to PEMDAS.
2. x + xy = 1
   The thread title should really be "Find solutions of x + xy = 1", since the question is more about graphing this equation than adding some numbers. This is more complicated because now the result of adding x and xy must also be equal to 1. This is an equation, so the goal is to find pairs of numbers, x and y, so that the sum on the left side equals 1. As already mentioned, the way to do this is to solve the equation for one variable (solving for y is easier in this case), and then using that new equation to find pairs of numbers that are solutions.
   Solving for y here yields y = 1/x - 1, so some pairs of numbers that are solutions are (1, 0), (2, -1/2), (3, -2/3), and so on. There are an infinite number of solutions, all of which make up the graph shown in post #4.

 

[pbuk]

So, if I have to build complex equation that is in 'x' and 'y'

The word 'complex' has a special meaning in mathematics which makes it inappropriate here: the closest alternative would be 'complicated'.

then I need to gather data points that has both 'x' and 'y'. Then build equation? How?

It is not in general a worthwhile exercise to try to do this. For instance here are five curves that all pass through the points (-1, 1), (0, 0) and (1, 1): https://www.desmos.com/calculator/qll5z99rjo. How would you know which one you want?

I need to try to fit the values into some curve? How? Interpolation?

Yes, if you want a smooth curve that joins up a series of points then some form of interpolation e.g. cubic spline interpolation is the way to go. Although it may seem like this is a similar topic to sketching the curve of a known function this is not actually the case.

 

[pairofstrings]

This is more complicated because now the result of adding x and xy must also be equal to 1. This is an equation, so the goal is to find pairs of numbers, x and y, so that the sum on the left side equals 1. As already mentioned, the way to do this is to solve the equation for one variable (solving for y is easier in this case), and then using that new equation to find pairs of numbers that are solutions.

Assume that I am trying to build a heart curve then how do I find out if the equation needs to have a 'x' raised to the power of '6', how to find out if the equation needs to have 3$x^{4}y^{2}$.. as one of its terms?

Heart curve equation looks like this:

The thread's first post was a heart curve simplified to x + xy = 1 to find out how I can add curve like $x^{6}$ to curve 3$x^{4}y^{2}$.

Thanks.

 

[pbuk]

You need to stop obsessing about this 'heart curve'. This is nothing more than an amusing curiosity probably produced after a lot of trial and error by some graduate student in order to impress someone around St. Valentine's Day.

 

[pairofstrings]

You need to stop obsessing about this 'heart curve'.

I don't have any other equation to study.
I am studying it because it is popular among most of the people.
Heart curve has more resources on the web that could help me understand equations.

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I am able to understand simple polynomial equations like Monomial, Binomial, Trinomial, Quadratics, Cubic. When I say "I understand" I am saying that I am familiar with how these equations look like.

But to understand them I need to know how the terms of an equation impacts other terms of the equation when used with plus or minus between terms, which you say is not the way to understand graphs.

Terms like $x^{6}$ when equated to one looks like this:

Then if I add another term: $3x^{4}y^{2}$ to $x^{6}$

The following is what I get:

That is why I am asking that how $x^{6}$ impacts $3x^{4}y^{2}$.

..an amusing curiosity probably produced after a lot of trial and error..

How do I do lot of 'trial and error' to get a curve?
What is lot of 'trial and error'?
Thanks.

 

[Gaussian97]

I would recommend you to look at this video:

It covers the basic tools to "draw" using equations.

This method has the problem that has a lot of points where the functions are not differentiable, so don't look smooth. But I think it should be possible to draw easy curves like the heart one with a simple method, for example:
Draw by hand the curve you want and find a number of points (say 1000) of the curve. You want an equation $f(x,y)=0$ that matches your curve, start with an ansatz for $f$ with some parameters (you can start with polynomial ansatz, like the heart curve), then use any method to fit the parameters to minimize the distance to the 1000 points you have.
With this you should be able to find the heart curve and a lot more, the more complex your ansatz is, the more complex curves you will be able to reproduce.
That's just a first idea, maybe doesn't work properly, but sure enough, there will be more complex algorithms to find smooth equations like the heart one.

 

[Mark44]

Thread locked.

Assume that I am trying to build a heart curve then how do I find out if the equation needs to have a 'x' raised to the power of '6', how to find out if the equation needs to have

Enough with this heart curve. When you are able to understand very simple graphs, such as linear, quadratic, square root and a few others, then, and only then, will we entertain a discussion of the heart curve

You need to stop obsessing about this 'heart curve'.

Amen!

I don't have any other equation to study.

Get yourself a precalculus textbook. It will have lots of equations whose features you can study.