System of Eqns (Containing Absolute Value Operators): 0.25 = |P + Q|, 1 = |P| + |Q|

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tempusfugitful

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Good Day All,

I have a system of equations here that I am having difficulty resolving. The absolute value signs are giving me grief and I can't figure out how to isolate the variables.

Suppose the following.

.25 = |P + Q|
1 = |P| + |Q|

Would someone please show me the derivation to solve for P and Q?

Sincerely,
Tempus
 
tempusfugitful said:
Hello Everyone,

I cannot figure out a way to solve for P and Q in the following two equations.

.25 = |P + Q|
1.0 = |P| + |Q|

Any ideas?
Click to expand...
Plot those functions.

Look at the point of intersections - that will give you clues to method/s of solutions.
 
Are you saying that the equations cannot be solved with algebraic methods, but must rather be analyzed graphically?
 
tempusfugitful said:
Good Day All,

I have a system of equations here that I am having difficulty resolving. The absolute value signs are giving me grief and I can't figure out how to isolate the variables.

Suppose the following.

.25 = |P + Q|
1 = |P| + |Q|

Would someone please show me the derivation to solve for P and Q?

Sincerely,
Tempus
Click to expand...
Hi, IMO to do this problem you need to understand absolute values. For instance if |p+q|-.25, then what value or values, if any, can p+q equal?

There are four cases for p and q-- p>0 and q>0 OR p>0 and q<0 ,,,, What affect does these 4 cases have on |p|+|q|=1???
 
Subhotosh Khan said:
Plot those functions.

Look at the point of intersections - that will give you clues to method/s of solutions.
Click to expand...

I am saying - when you have no idea about solving a set of equations - graphing those is a fruitful first step.

That will let you know the number of solutions (in this case 4) and also guide you regarding the method of solving those.
 
Jomo said:
Hi, IMO to do this problem you need to understand absolute values. For instance if |p+q|-.25, then what value or values, if any, can p+q equal?

There are four cases for p and q-- p>0 and q>0 OR p>0 and q<0 ,,,, What affect does these 4 cases have on |p|+|q|=1???
Click to expand...

I understand what you are saying, and I have done the graphing analysis. This feels like a heuristic method to me. What I was hoping for was some sort of matrix-algebra style solution to a system of equations. In your judgement, this is not possible due to the absolute value operators?
 
tempusfugitful said:
I have a system of equations here that I am having difficulty resolving. The absolute value signs are giving me grief and I can't figure out how to isolate the variables.

Suppose the following.

0.25 = |P + Q|
1 = |P| + |Q|

Would someone please show me the derivation to solve for P and Q?
Click to expand...
Start with the definition of "absolute value", and go from there.

In your case, |P + Q| = 0.25 when P + Q = 0.25 or P + Q = -0.25. These solve as P = 0.25 - Q or P = -0.25 - Q. The other equation can be extracted as 1 = P + Q (for P > 0 and Q > 0), 1 = -P + Q (for P < 0 and Q > 0), 1 = P - Q (for P > 0 and Q < 0), and 1 = -P - Q (for P < 0 and Q < 0).

Consider each of the second equation's four cases separately, using substitution from the first equation. So, for example:

. . .P < 0 and Q < 0, with P + Q = 0.25:
. . . . .1 = -P - Q = -(0.25 - Q) - Q
. . . . .1 = -0.25 + Q - Q = -0.25

But 1 does not equal -0.25, so this set of inputs has no solution.

. . .P < 0 and Q < 0, with P + Q = -0.25:
. . . . .1 = -P - Q = -(-0.25 - Q) - Q
. . . . .1 = 0.25 - 2Q
. . . . .0.75 = -2Q
. . . . .-3/8 = Q
. . . . .P = -1/4 + 3/8 = 1/8

However, P cannot be both less than zero and also a positive number, so this set of inputs also has no solution.

Now consider each of the other three cases. ;)
 
tempusfugitful said:
I understand what you are saying, and I have done the graphing analysis. This feels like a heuristic method to me. What I was hoping for was some sort of matrix-algebra style solution to a system of equations. In your judgement, this is not possible due to the absolute value operators?
Click to expand...
Two cases for the first equality, (P+Q)\(\displaystyle \le\)0 and P+Q\(\displaystyle \ge\)0, and 4 for the second, P and Q \(\displaystyle \le\) 0, etc. So, 2*4=8 cases.

Some are redundant or inconsistent (actually 4 of them) and 4 lead to values for P and Q.

I find graphing simpler in this case.
 
Thank you all for the help. I haven't done this sort of math in ages. You helped me knock the cobwebs out of my head on this one.

Also, can someone tell me how one can post math equations inside the text? Must I use a markup like LaTeX or something?
 
tempusfugitful said:
Thank you all for the help. I haven't done this sort of math in ages. You helped me knock the cobwebs out of my head on this one.

Also, can someone tell me how one can post math equations inside the text? Must I use a markup like LaTeX or something?
Click to expand...
The particular tag pair used here to enclose the LaTex forms is [t e x] [/t e x] without the spaces. You can see samples by doing a reply to a post which has formula's, etc. For example
Code: 
[tex]\begin{pmatrix} a & b \\ c & d \end{pmatrix}[/tex]

[tex]\sqrt{\dfrac{1+x^2}{4}}\, =\, \dfrac{\sqrt{1+x^2}}{2}[/tex]
Produces
\(\displaystyle \begin{pmatrix} a & b \\ c & d \end{pmatrix}\)

\(\displaystyle \sqrt{\dfrac{1+x^2}{4}}\, =\, \dfrac{\sqrt{1+x^2}}{2}\)




Among others, a reference I use for the actual LaTex is
http://tex.wikidot.com/start
 
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