Solving System of Equations: xy, yz, zx

In summary, a system of equations is a set of equations with multiple variables that must be solved simultaneously. This can be done using methods like substitution, elimination, or graphing. Solving a system of equations is important because it allows for the solution of real-world problems in various fields. Linear systems have equations in the form of y = mx + b, while non-linear systems have at least one equation that cannot be written in this form. A system of equations can have any number of variables, but the number of equations must be equal to or greater than the number of variables in order to solve the system.
  • #1
solakis1
422
0
Solve the following systemof equations:

$\dfrac{xy}{x+y}=a$

$\dfrac{yz}{y+z}=b$

$\dfrac{zx}{z+x}=c$

where a,b,c are not zero
 
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  • #2
solakis said:
Solve the following systemof equations:

$\dfrac{xy}{x+y}=a$

$\dfrac{yz}{y+z}=b$

$\dfrac{zx}{z+x}=c$

where a,b,c are not zero
Inverting the 3 we get$\frac{1}{y} + \frac{1}{x} = \frac{1}{a}\dots(1)$$\frac{1}{y} + \frac{1}{z} = \frac{1}{b}\dots(2)$$\frac{1}{z} + \frac{1}{x} = \frac{1}{c}\dots(3)$Add the 3 to get$2(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$Or $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{2}(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$Subtracting (1) from above we get$\frac{1}{z} = \frac{1}{2}(\frac{1}{b} + \frac{1}{c}- \frac{1}{a})$Or $\frac{1}{z} = \frac{1}{2}\frac{ac + ab - bc}{abc}$Or $z= \frac{2abc}{ac + ab - bc}$ Similarly $x= \frac{2abc}{ab + bc - ac}$And $y= \frac{2abc}{ac - ab + bc}$
 
  • #3
nery good
 
  • #4
solakis said:
very good
 

1. What is a system of equations?

A system of equations is a set of two or more equations that contain multiple variables. The goal is to find the values of the variables that satisfy all of the equations simultaneously.

2. How do you solve a system of equations?

The most common method for solving a system of equations is by substitution or elimination. In substitution, one variable is isolated in one equation and then substituted into the other equation. In elimination, one variable is eliminated by adding or subtracting the equations together.

3. What is the importance of solving systems of equations?

Solving systems of equations is important in many fields of science, such as physics, chemistry, and engineering. It allows us to find the relationships between multiple variables and make predictions about how they will behave in different situations.

4. Can a system of equations have more than one solution?

Yes, a system of equations can have one, infinite, or no solutions. If the equations are consistent and independent, there will be one unique solution. If the equations are consistent and dependent, there will be an infinite number of solutions. If the equations are inconsistent, there will be no solutions.

5. How do you know if a system of equations has no solution?

If the system of equations results in a contradiction, such as 0=3, then there is no solution. This means that the equations are inconsistent and there is no set of values that can satisfy all of the equations simultaneously.

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