Disgusting Algebra, is this solvable by hand?

In summary, the conversation discusses solving an equation using a CAS and whether it can be solved by hand. The problem involves a 2 dimensional GPS and the use of the distance formula. Various methods were tried, including substitution and factoring, but they all resulted in a messy solution. The conversation also mentions using linear algebra and redefining constants as possible solutions. Ultimately, the conversation ends with the need for back substitution and the acknowledgement that it will be a messy process.
  • #1
DrummingAtom
659
2
When do I resort to using a CAS to solve an equation? Is this one possible by hand?

I got this from a 2 dimensional GPS problem. Here's the math using the distance formula.

L12 = (x1- x)2 + (y1- y)2

L22 = (x2- x)2 + (y2- y)2

I tried the substitution method but it gets nasty and I would have 4th power equations. I don't know if there is a clean answer because Wolfram gave me a very long answer. I also used Wolfram to calculate the numerical answer. Thanks for any help.
 
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  • #2
Ooops.. Solving for x and y.
 
  • #3
The first thing that came to mind was calculate L1-L2 and factorize.
 
  • #4
Yeah I did try that too but it still needs a back substitute that uses a very long chain of squared terms. I guess I'm just hoping there is a more elegant method because I don't know much about Linear Algebra.
 
  • #5
DrummingAtom said:
Yeah I did try that too but it still needs a back substitute that uses a very long chain of squared terms.

What? Please post your working so far.
 
  • #6
L12 = (x1- x)2 + (y1- y)2

L22 = (x2- x)2 + (y2- y)2

Then L12 - L22 and factor:

L12 - L22 = (x12 - 2xx1 + x2) - (y12 - 2yy1 + y2) - (x22 - 2xx2 + x2) + (y22 - 2yy2 + y2)

Some algebra:

x = [L12 - L22 - (x12 + y12 + x22 - y22 - 2yy1 + 2yy2]/(-2x1 + 2x2)

Now I have to back sub to one of the original equations.. That's going to be a mess.
 
  • #7
Set:

x=Ay+B.

Then it won't be so messy at all.

Determine what A and B is later on, (once you've solved for y and x) from your last line given in your post.
 
  • #8
DrummingAtom said:
Then L12 - L22 and factor:

Well you expanded, not factorized. I was thinking using to difference of two squares to obtain
L12-L22 = (x1+x2-2x)(x1-x2) + (y1+y2-2y)(y1-y2)
which is just a linear equation. As arildno suggested, redefine the constants.

Now I have to back sub to one of the original equations.. That's going to be a mess.

Why do you need to do that?
 
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What is "Disgusting Algebra"?

"Disgusting Algebra" is a term used to describe complex and challenging algebraic equations or problems that may seem overwhelming or unpleasant to solve.

Can "Disgusting Algebra" be solved by hand?

Yes, "Disgusting Algebra" can be solved by hand. However, it may require a lot of time, effort, and advanced mathematical skills.

What are some tips for solving "Disgusting Algebra" problems?

Some tips for solving "Disgusting Algebra" problems include breaking down the problem into smaller, more manageable steps, using algebraic rules and properties, and checking your work carefully.

Is it necessary to solve "Disgusting Algebra" problems by hand?

No, it is not necessary to solve "Disgusting Algebra" problems by hand. There are many online tools and calculators available that can help to solve these types of problems.

How can I improve my skills in solving "Disgusting Algebra" problems?

To improve your skills in solving "Disgusting Algebra" problems, it is important to practice regularly, seek help from teachers or tutors, and familiarize yourself with various algebraic techniques and strategies.

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