Solve for $a+b+c$: Equation System ($a,b,c\in N$)

In summary, an equation system is a set of two or more equations that are solved simultaneously to find values of the variables involved. The notation $a,b,c\in N$ indicates that the variables a, b, and c are natural numbers. To solve for $a+b+c$, each equation in the system must be solved to find the values of a, b, and c, which are then added together. This process applies even if there are more than three variables in the system. Additionally, an equation system can have multiple solutions, meaning there can be more than one combination of values that satisfy all of the equations.
  • #1
Albert1
1,221
0
$a,b,c \in N$, and the following equation system is given :

$\left\{\begin{matrix}
ab+bc+ca+2(a+b+c)=8045-----(1) & & & & \\
abc-a-b-c=-2-----(2) & & & &
\end{matrix}\right.$

find the value of $a+b+c$
 
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  • #2
My solution:

Note that $(a+1)(b+1)(c+1)=abc+ab+bc+ca+a+b+c+1$. Now, substitute what we're given into it, we get $(a+1)(b+1)(c+1)=-2+8045+1=8044=2(2)(2011)$, this implies $(a,\,b,\,c)=(1,\,1,\,2010)$ (up to permutations) and hence $a+b+c=2012$.
 

FAQ: Solve for $a+b+c$: Equation System ($a,b,c\in N$)

1. What is an equation system?

An equation system is a set of two or more equations that are to be solved simultaneously in order to find the values of the variables involved.

2. What does $a,b,c\in N$ mean?

The notation $a,b,c\in N$ means that the variables a, b, and c are all natural numbers, which are positive integers (whole numbers greater than 0).

3. How do I solve for $a+b+c$ in an equation system?

To solve for $a+b+c$, you must first solve each individual equation in the system to find the values of a, b, and c. Then, simply add the values together to find the sum.

4. What if there are more than three variables in the equation system?

If there are more than three variables in the equation system, the same process applies. You must solve each equation to find the values of the variables, and then add them together to find the sum.

5. Can there be more than one solution to an equation system?

Yes, an equation system can have multiple solutions. This means that there can be more than one combination of values for the variables that satisfy all of the equations in the system.

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