How are these two equations the same?

Thread starter IntegrateMe
Start date Apr 15, 2012

In summary, two equations can be considered the same if they have the same solution set and can be transformed into each other using algebraic operations. To prove equivalence, one can show that both equations have the same values for the variables or manipulate one equation into the other. Different forms of equations can also be considered the same if they have the same solution set. Common ways to show equivalence include solving both equations, graphing them, or using algebraic operations.

Apr 15, 2012

IntegrateMe

[tex]\lim_{n\to\infty}\frac{8^{n-1}}{9^{n}}[/tex]

and

[tex]\frac{1}{9}\lim_{n\to\infty}(\frac{8}{9})^{n}[/tex]

In particular, why is there a 1/9 beside the limit instead of a 1/8 in the second equation?

Last edited: Apr 15, 2012

Physics news on Phys.org

Apr 15, 2012

Poopsilon

These are not equations, FYI, they are limits, and as limits they are the same.

1. How can two equations be considered the same?

Two equations can be considered the same if they have the same solution set. This means that both equations have the same values for the variables that make them true. For example, the equations 2x + 3 = 7 and x = 2 are considered the same because both have the solution x = 2.

2. What makes two equations equivalent?

Two equations are equivalent if they have the same solution set and can be transformed into each other using algebraic operations. This means that if you apply the same operations to both equations, you will get the same result. For example, the equations x + 2 = 5 and x = 3 are equivalent because you can subtract 2 from both sides of the first equation to get the second equation.

3. How do you prove that two equations are the same?

To prove that two equations are the same, you need to show that they have the same solution set. This can be done by solving both equations and showing that they have the same values for the variables. You can also prove equivalence by manipulating one equation into the other using algebraic operations.

4. Can two equations with different forms be considered the same?

Yes, two equations with different forms can still be considered the same if they have the same solution set. For example, the equations x^2 + 4x + 4 = 0 and (x + 2)^2 = 0 are considered the same because they both have the solution x = -2.

5. What are some common ways to show that two equations are the same?

Some common ways to show that two equations are the same include solving both equations and showing that they have the same values for the variables, graphing both equations and showing that they have the same points of intersection, or manipulating one equation into the other using algebraic operations.