Solve for x

[qwertysaur]

Incorrect. If you substitute x = 0 then you will get -7 = 19 which is wrong. The correct answer is no solution.
Proof
<pre>
x-7 = 19 + x
-x -x</pre>
-7 = 19, a contradiction indicating no solution.
You can see the fallacy in your work because ito get x=0 you had to multiply everything by x. However assume x is 0 you are simply multiplying everything by zero and all information is lost.

but I added and mutiplied everything on both sides. Ergo it is valid. x = 0

If you want to prove that x = 0, then you can't multiply both sides of the equation by x to get that solution, because it will always result in 0 = 0. (Side note you also can't divide by x to prove x = 0 because dividing by zero results in no solution.) Take for example x+1 = 2. It should be obvious that x = 1, but lets go with pg's method. Multiply everything by x to get x²+x = 2x. Subtract 2x from both sides and you get x²-x = 0. Factor and you have (x)(x-1) = 0, giving the solutions x=0 and x=1. However, if x=0 then you get 1 = 2 which is a contradiction.


There are many problems where you have to reject a solution because an answer makes no sense. For example you have a rectangle where the length of one side is three units less than that of the other, and the area is found to be 4 square units. You get (x)(x-3) = 4, which becomes x²-3x = 4. Subtract 4 form both sides to get x²-3x-4 = 0. Factor to get (x-4)(x+1) = 0. So x = 4 or x = -1. You have to reject x = -1 because you can't have a side with negative length.

Remember to always check to make sure your solutions actually make sense. Also when using an esoteric process for finding the solution, make sure there is no way that is easier first. Don't use calculus to find the area of a circle, just use A = πr². The calculus method is possible, but much more time consuming.