Can You Solve This Week's Challenging System of Equations?

  • MHB
  • Thread starter anemone
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In summary, a system of equations is a set of two or more equations with multiple variables that are solved simultaneously. There are various methods for solving these equations, including substitution and elimination. A system of equations can have any number of variables, but the number of equations must be equal to the number of variables for a unique solution. Consistent systems have at least one solution while inconsistent systems have no solution. Systems of equations are used in various fields to model and solve real-world problems.
  • #1
anemone
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MHB
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Here is this week's POTW:

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Solve the system of equation

$3x+7y+14z=252\\xyz-u^2=2016$

for non-negative real numbers.

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  • #2
No one answered last week's POTW. (Sadface) However, you can find the suggested model solution as follows:
From the conditions of the problem we obtain

$\begin{align*} 252=3x+7y+14z& \ge 3\sqrt[3]{3x(7y)(14z)}\\&=3\sqrt[3]{3(7)(14)(2016+u^2)}\\& \ge 3\sqrt[3]{3(7)(14)(2016)}\\&=3\sqrt[3]{2^6(3^3)(7^3)}\\&=2^2(3^2)(7)\\&=252\end{align*}$

Equality is attainable when $3x=7y=14z$ and $u=0$. From the first equation of the system, we obtain

$3x=7y=14z=\dfrac{252}{3}=84$. This implies $x=28,\,y=12$ and $z=6$.
 

FAQ: Can You Solve This Week's Challenging System of Equations?

What is a system of equations?

A system of equations is a set of two or more equations with multiple variables that are to be solved simultaneously. The solution to a system of equations is a set of values for each variable that satisfies all of the equations in the system.

How do you solve a system of equations?

There are several methods for solving a system of equations, including substitution, elimination, and graphing. The most common method is substitution, where one equation is solved for one variable and then substituted into the other equation. This process is repeated until all variables have been eliminated, resulting in a solution for the remaining variable.

What is the importance of systems of equations in science?

Systems of equations are used in science to model and solve real-world problems. They can be used to represent relationships between different variables and predict outcomes in various scientific fields, such as physics, chemistry, and biology.

Can systems of equations have more than two variables?

Yes, systems of equations can have any number of variables. However, the number of equations in the system must be equal to the number of variables in order for a unique solution to exist. If there are more variables than equations, the system is considered underdetermined and will have an infinite number of solutions.

How can I check if my solution to a system of equations is correct?

You can check your solution by substituting the values into each equation in the system. If the values satisfy all of the equations, then your solution is correct. You can also graph the equations and see if the point of intersection matches your solution.

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