# General solution of a system of equations and partial fractions

• phy$x In summary, the conversation discusses solving for a system of equations and finding a general solution, as well as expressing a complex expression in partial fractions. The conversation also includes a clarification on notation and the use of parentheses. phy$x
I've been trying to get out this question for a while now:

ai) Show that (x,y,z) = (1,1,1) is a solution to the following system of equations:

x + y + z = 3
2x + 2y + 2z = 6
3x + 3y +3z = 9

aii) Hence find the general solution of the system

b) Express 2x^2 + 3/(x^2 + 1)^2 in partial fractions

My attempt:

Well ai) was simple and i got that part out with barely any effort.

In aii), i don't even know how to start :( All i know is that the answer is supposed to be:

(x,y,x) = λ(1,0,-1) + μ(0,1,-1) + (1,1,1,)

Sorry i can't offer any attempt...i just really don't know where to start...any help at all will be appreciated here.

With b) i used the matrix method...but that wasnt the approach they were looking for. I was supposed to use the concept of repeated factors:

2x^2 + 3/(x^2 + 1)^2 = Ax + B/x^2 + 1 + CX + D/(x^2+1)^2
(multiply throughout by (x^2 + 1)^2)
2x^2 +3 = (Ax + B)(x^2 + 1) + Cx + D
Let x=0
3 = B + D
D = 3 - B
Let x= 1
5 = (A + B)(2) + C + D
5 = 2A + 2B + C + D

Substituting D = 3 - B

5 = 2A + 2B + C + 3 - B
2 = 2A + B + C

Well this is where I am stuck...any help at all would be a life saver. Thank you in advance.

Strange question. Clearly all three equations are equivalent, so we can ignore the 2nd and 3rd.
Since x+y+z is linear, having found a solution of x+y+z=3, we can add to it any solution of x+y+z=0 and the result will be a solution of the original equation. So the question is essentially asking you to find all solutions of x+y+z=0.
2x^2 + 3/(x^2 + 1)^2
Seems you mean (2x2 + 3)/(x2 + 1)2. Please use parentheses properly and subscript/superscript. Makes expressions much more readable.
2x^2 +3 = (Ax + B)(x^2 + 1) + Cx + D
The usual procedure from this point is to separate out each power of x into a different equation. That will give you four equations here.

phy$x said: I've been trying to get out this question for a while now: ai) Show that (x,y,z) = (1,1,1) is a solution to the following system of equations: x + y + z = 3 2x + 2y + 2z = 6 3x + 3y +3z = 9 aii) Hence find the general solution of the system b) Express 2x^2 + 3/(x^2 + 1)^2 in partial fractions My attempt: Well ai) was simple and i got that part out with barely any effort. In aii), i don't even know how to start :( All i know is that the answer is supposed to be: (x,y,x) = λ(1,0,-1) + μ(0,1,-1) + (1,1,1,) Sorry i can't offer any attempt...i just really don't know where to start...any help at all will be appreciated here. With b) i used the matrix method...but that wasnt the approach they were looking for. I was supposed to use the concept of repeated factors: 2x^2 + 3/(x^2 + 1)^2 = Ax + B/x^2 + 1 + CX + D/(x^2+1)^2 (multiply throughout by (x^2 + 1)^2) 2x^2 +3 = (Ax + B)(x^2 + 1) + Cx + D Let x=0 3 = B + D D = 3 - B Let x= 1 5 = (A + B)(2) + C + D 5 = 2A + 2B + C + D Substituting D = 3 - B 5 = 2A + 2B + C + 3 - B 2 = 2A + B + C Well this is where I am stuck...any help at all would be a life saver. Thank you in advance. In (b), do you mean $$2x^2 + \frac{3}{(x^2+1)^2}$$ (which is what you wrote), or do you mean $$\frac{2x^2 + 3}{(x^2 + 1)^2)}?$$ If you meant the former, then what you wrote is perfectly OK, but if you mean the latter, you must use parentheses, like this: (2x^2 + 3)/(x^2+1)^2. Using ASCII and hence things like x^2 is OK, but you must write clearly. Also, where you write Ax + B/x^2 + 1 + CX + D/(x^2+1)^2, you are writing $$A x + \frac{B}{x^2} + 1 + Cx + \frac{D}{(x^2+1)^2}.$$ I hope that is not what you really mean, but again, without parentheses, your expressions are impossible to parse (and, frankly, it takes too much of my time, so I won't even try). i meant: (Ax + B)/(x^2 + 1) + (Cx + D)/((x^2 + 1)^2) Sry about that... phy$x said:
I've been trying to get out this question for a while now:

b) Express 2x^2 + 3/(x^2 + 1)^2 in partial fractions

My attempt:

With b) i used the matrix method...but that wasn't the approach they were looking for. I was supposed to use the concept of repeated factors:

2x^2 + 3/(x^2 + 1)^2 = Ax + B/x^2 + 1 + CX + D/(x^2+1)^2
(multiply throughout by (x^2 + 1)^2)
2x^2 +3 = (Ax + B)(x^2 + 1) + Cx + D
Let x=0
3 = B + D
D = 3 - B
Let x= 1
5 = (A + B)(2) + C + D
5 = 2A + 2B + C + D

Substituting D = 3 - B

5 = 2A + 2B + C + 3 - B
2 = 2A + B + C

Well this is where I'm stuck...any help at all would be a life saver. Thank you in advance.
Expanding the product and collecting terms in the equation

2x2 +3 = (Ax + B)(x2 + 1) + Cx + D

gives

$\displaystyle 2x^2+3=Ax^3+Bx^2+(A+C)x+(B+D)\ .$

Now equate coefficients of each power of x.

Thanks a lot guys! All help greatly appreciated. :)

## 1. What is the general solution of a system of equations?

The general solution of a system of equations is the set of all possible solutions that satisfy all of the given equations in the system. It can be represented as a set of equations or as a set of values for the variables in the system.

## 2. How do you find the general solution of a system of equations?

To find the general solution of a system of equations, you can use different methods such as substitution, elimination, or graphing. These methods involve manipulating the equations in the system to isolate the variables and solve for their values.

## 3. What are partial fractions?

Partial fractions are a method for breaking down a rational function into simpler fractions. It involves breaking down the denominator into linear or quadratic factors and finding the coefficients for each factor.

## 4. How are partial fractions used in finding the general solution of a system of equations?

Partial fractions can be used to solve systems of equations when the equations involve rational functions. By breaking down the rational functions into simpler fractions, the equations can be solved for the values of the variables.

## 5. Are there any limitations to using partial fractions in finding the general solution of a system of equations?

Yes, there are limitations to using partial fractions. This method can only be used for systems of equations that involve rational functions. It may also not be applicable in cases where the factors in the denominator are not distinct or when the degree of the numerator is greater than or equal to the degree of the denominator.

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