How are these two equal?(equation, inequality)

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In summary, the conversation discusses a problem in discrete mathematics involving finding the number of integer solutions for two equations, one with a less than sign and one with an equal sign. The solution to the problem is explained as being a result of being able to add a positive number to the solutions of the first equation to make it equal to the second equation. This means that both equations have the same number of integer solutions.
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How are these two equal??(equation, inequality)

I study discrete mathematics and we are doing combinations at the moment. There is this example in the book(Discrete Mathematics and Combinatorics p. 30 Ex. 1.43) where it states that the number of integer solutions for:

x1+x2+x3+x4+x5+x6<10 where xi[tex]\geq[/tex]0

is equal to the number of integer solutions of

x1+x2+x3+x4+x5+x6+x7=10 where xi[tex]\geq[/tex]0 and x7>0

Can someone explain me this? The author supposes that I magically understand what goes through his mind.

Here is a screenshot of the problem: http://img251.imageshack.us/img251/823/garbageab.jpg [Broken]
 
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This is subtle, but, trivial: Try to see that to all the possbile solutions of x1+x2+x3+x4+x5+x6<10, we can add a positive number and make it equal to 10 and also that it is only the set of these solutions to which we can add a positive number and make it equal to 10; thus the set of integer solutions of both equations have the same number of elements.
 

1. How do I know if two equations or inequalities are equal?

Two equations or inequalities are considered equal if they have the same solution set. This means that if you plug in the same values for the variables in both equations, you will get the same result. Additionally, if you graph the equations, they will produce the same line or shape.

2. Can two equations or inequalities with different forms be equal?

Yes, two equations or inequalities with different forms can still be equal if they have the same solution set. For example, the equations y = 2x + 3 and 2x - y = -3 are different forms but have the same solution set of (x, y) = (0, 3).

3. How can I prove that two equations or inequalities are equal?

To prove that two equations or inequalities are equal, you can use algebraic manipulation. By performing the same operations on both equations or inequalities, you can show that they are equivalent and have the same solution set. You can also use graphs to visually demonstrate that the two equations or inequalities produce the same line or shape.

4. Are all equations or inequalities equal to themselves?

Yes, all equations or inequalities are equal to themselves. This is because any value that satisfies the equation or inequality will also satisfy itself. For example, the equation x + 2 = 5 is equal to itself because x = 3 satisfies both sides of the equation.

5. Can an equation or inequality with no solution be equal to another equation or inequality?

No, an equation or inequality with no solution cannot be equal to another equation or inequality. This is because they do not have the same solution set. For example, the equations x + 2 = 5 and x + 2 = 6 have different solution sets and are not considered equal.

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