Solve System of n Equations: Can You Help?

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In summary, to solve a system of equations, you can use methods such as substitution, elimination, or graphing. A system of equations refers to a set of two or more equations with multiple variables that can be solved simultaneously. 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 to find a solution. If the system of equations has no solution, it means the equations are inconsistent. There is no one shortcut to solving all systems of equations, but there are strategies and techniques that can make the process more efficient.
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Mathick
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Solve the system of \(\displaystyle n\) equations with unknown \(\displaystyle x_1, x_2, ... , x_n\), for \(\displaystyle n\ge 2\):

\(\displaystyle 2x_1^3 + 4 =x_1^2 (x_2 +3)\)
\(\displaystyle 2x_2^3 + 4 =x_2^2 (x_3 +3)\)
\(\displaystyle ......\)
\(\displaystyle 2x\)n-13\(\displaystyle + 4 = \)\(\displaystyle x\)n-12\(\displaystyle (x_n +3)\)
\(\displaystyle 2x_n^3 + 4 =x_n^2 (x_1 +3)\)

Can you help me?
 
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The symmetry of the equations led me to wonder what would happen if all the variables were equal. In that case, there are two solutions. $x_j=2$ is a double-root, and $x_j=-1$ is a single root. I don't think there is any guarantee that these are the only solutions, but you can see by inspection that they both solve the system.
 

FAQ: Solve System of n Equations: Can You Help?

How do you solve a system of equations?

To solve a system of equations, you can use various methods such as substitution, elimination, or graphing. The method you use will depend on the specific equations and variables involved.

What is a system of equations?

A system of equations refers to a set of two or more equations that contain multiple variables. The equations are related and can be solved simultaneously to find the values of the variables that satisfy all equations.

Can you solve a system of equations with more than two variables?

Yes, a system of equations can have any number of variables. However, the number of equations must be equal to or greater than the number of variables in order to solve the system.

What if the system of equations has no solution?

If a system of equations has no solution, it means that the equations are inconsistent and do not intersect at any point. This could happen if the equations are parallel or if they represent the same line.

Is there a shortcut to solving systems of equations?

There is no one shortcut to solving all systems of equations. However, there are some strategies and techniques that can make the process more efficient, such as using matrices or substitution patterns.

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