Solving equations but getting an identity - what mistakes were made?

[farfromdaijoubu]

TL;DR

What are the mistakes made when you try to solve equations for some variable/s but end up with an identity?

This might be a bit vague but when solving algebra equations, what does it 'mean', or what mistakes does it imply if you end up with both sides of the equation being the same thing and getting nowhere? For example, you want to solve a system for x, but the x's end up cancelling and you get 0=0.

For context I was just doing some basic projectile motion stuff - wanted to find minimum speed at which a ball thrown at a given angle would always collide with another dropped at the same instant but ended up constant = constant.

But I've run into the same issue many times before and never properly learnt what I was doing wrong whenever it happened.

 

[fresh_42]

It usually means that your variable is actually a degree of freedom and can be set to anything. However, it could also mean that you made a mistake, like using the statement that you want to show, or simply typos. It depends on the case.

 

[PeroK]

TL;DR Summary: What are the mistakes made when you try to solve equations for some variable/s but end up with an identity?

This might be a bit vague but when solving algebra equations, what does it 'mean', or what mistakes does it imply if you end up with both sides of the equation being the same thing and getting nowhere? For example, you want to solve a system for x, but the x's end up cancelling and you get 0=0.

For context I was just doing some basic projectile motion stuff - wanted to find minimum speed at which a ball thrown at a given angle would always collide with another dropped at the same instant but ended up constant = constant.

But I've run into the same issue many times before and never properly learnt what I was doing wrong whenever it happened.

You don't need to do anything wrong to get an identity. An identity is a true statement. It just means that you didn't end up where you wanted to.

For example, if you use integration by parts twice, changing the roles of the functions, you tend to end up with an identity.

 

[Mark44]

What are the mistakes made when you try to solve equations for some variable/s but end up with an identity?

There are three kinds of equations: conditional equations, identities, and contradictions.
A conditional equation is one in which the variable can take on only a limited number of values for the equation to be a true statement. For example, $x^2 - 3x + 2 = 0$. This equation is a true statement only for x = 1 or x = 2.

An identity is an equation that is a true statement for all values of the variable. For example, $x^2 - 3x + 2 = (x -1)(x - 2)$. Any real value (or even a complex value) can be substituted for x here and the resulting equation will be a true statement.

A contradiction is an equation that can never be true, no matter which value the variable takes on. For example, $x = x + 1$ is never true for any value.

This might be a bit vague but when solving algebra equations, what does it 'mean', or what mistakes does it imply if you end up with both sides of the equation being the same thing and getting nowhere? For example, you want to solve a system for x, but the x's end up cancelling and you get 0=0.

It seems to me that you are starting from an equation that is an identity and applying operations to both sides to eventually get to 0 = 0. Can you give me a specific example of an equation that you're talking about?

 

[A.T.]

For context I was just doing some basic projectile motion stuff - wanted to find minimum speed at which a ball thrown at a given angle would always collide with another dropped at the same instant but ended up constant = constant.

For a system of equations this usually means that the equations are not independent of each other (redundant).