what does it mean when you are solving an equation, and, the variable that you are trying to solve for cancels itself out?
It means that there is no unique value for that variable. It could be anything and the equation would still be valid. Another way of saying this is: "The equation is an identity"
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When the variable that you are trying to solve for cancels itself out in an equation, it means that the equation has a special property called "identity" or "equality." This situation usually occurs when there is a common term or factor that can be divided or cancelled out on both sides of the equation.
To obtain a better understanding, let's walk through an example. Consider the equation:
3x + 6 = 9x + 6
Now, let's solve for the variable x:
First, we can simplify the equation by subtracting 6 from both sides:
3x = 9x
Next, we can subtract 3x from both sides:
0 = 6x
At this point, we can observe that 6x is multiplied by 0, resulting in 0 on the left side of the equation. This means that no matter what value x takes, the left side will always be 0. In other words, the equation is always true, and x itself doesn't affect the solution.
Hence, we say that the variable x has canceled itself out. The equation is an identity because it holds true for all values of x. In this case, we would say the solution is x = any real number, or "x is indeterminate" since it does not affect the equation.
When solving equations, it is essential to carefully analyze the steps taken to ensure no mistakes were made. If the variable cancels itself out, it's vital to verify that the resulting equation holds true for all possible values of the variable.