When rewrite and equation how do you know that your new equation is equivalent to your original equation?

Answer 1

There is no easy way to check this. You need practice in solving equations; anyway, here are a few considerations.

First of all, double-check whether you did everything correctly.If done correctly, some operations don't change the solution set of the original equation, i.e., they will give you an equivalent equation. This includes adding, subtracting, multiplying, and dividing both sides by the same number (but don't divide by zero!).

Multiplying both sides by a variable (or an expression that includes a variable) may add an additional solution. For example:

x = 5

has only one solution (5).

If you multiply both sides by "x", you get:

x^2 = 5x (x squared = 5x), which has the additional solution zero.

Similarly, dividing both sides of an equation by the same may eliminate a solution (as compared to the original equation); squaring may add solutions, and taking the square root may eliminate solutions.

Similarly, applying other functions, such as trigonometric functions and inverse trigonometric functions, may change the solution set - you need to be aware of the behavior of each specific type of function.