System of Equations: Find Real Numbers $p,q,r,s$

In summary, a system of equations is a set of two or more equations with multiple variables that have a set of values that satisfy all of the equations simultaneously. To solve a system of equations, there are different methods such as substitution, elimination, and graphing, with substitution being the most common. Real numbers are numbers that can be represented on a number line and are used in a system of equations to accurately represent quantities and relationships in real-world situations. To find real numbers in a system of equations, each variable is solved for in each equation, and the solutions are checked to see if they satisfy all of the equations in the system.
  • #1
anemone
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Find all real numbers $p,\,q,\,r,\,s$ that satisfy the following system of equations:

$spq+sp+pq+qs+s+p+q=9$

$rsp+rs+sp+pr+r+s+p=9$

$qrs+qr+rs+sq+q+r+s=9$

$pqr+pq+qr+rp+p+q+r=1$
 
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  • #2
anemone said:
Find all real numbers $p,\,q,\,r,\,s$ that satisfy the following system of equations:

$spq+sp+pq+qs+s+p+q=9$

$rsp+rs+sp+pr+r+s+p=9$

$qrs+qr+rs+sq+q+r+s=9$

$pqr+pq+qr+rp+p+q+r=1$

Add 1 to LHS and RHS of each expression to get
$(1+p)(1+q)(1+s) = 10$
$(1+p)(1+s)(1+r) = 10$
$(1+q)(1+r)(1+s) = 10$
$(1+p)(1+q)(1+r) = 2$

multiply all 3 and then take cube root to get
$(1+p)(1+q)(1+s)(1+r) = 10 \sqrt[3]{2}$

deviding above by 1st 3 equations
hence $(1+r)=(1+q)=(1+p) =\sqrt[3]{2}$

or $r=p=q=\sqrt[3]{2}-1$

and deviding by 4th equation we get

$s = 5\sqrt[3]{2}-1$
 

FAQ: System of Equations: Find Real Numbers $p,q,r,s$

What is a system of equations?

A system of equations is a set of two or more equations that have multiple variables, and the solution of the system is the set of values that satisfy all of the equations simultaneously.

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 of the equations is solved for one variable, and then that solution is substituted into the other equation to find the value of the other variable.

What are real numbers?

Real numbers are numbers that can be represented on a number line, including both positive and negative numbers, fractions, decimals, and integers. They are called "real" because they can represent physical quantities and measurements.

How do you find real numbers in a system of equations?

In a system of equations, real numbers can be found by solving for each variable in each equation, and then checking if the solutions satisfy all of the equations in the system. If a set of values satisfies all of the equations, then those values are considered real numbers in the system.

Why do we use real numbers in a system of equations?

We use real numbers in a system of equations because they accurately represent the quantities and relationships being studied. Real numbers can also be used to model real-world situations and solve problems in various fields such as science, engineering, and economics.

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