Question 12164
There are actually three types of equations to solve:

1.  When you solve for x and get one or more solutions this is called a "conditional equation."  This what usually happens, like when you solve x +3=5, the answer is x=2. This type of equation is called a CONDITIONAL EQUATION.

2.  Sometimes when you solve for x, it so happens that all the x's subtract out, and a false statement results.  For example, x+3 = 5 + x.  If you subtract x from each side, you get 3=5, which is NOT possible.  Therefore, there is NO SOLUTION, and that is where the answer is sometimes written with the greek letter phi (the circle with a line through it!). This type of equation is called a CONTRADICTION.


3.  Sometimes when you solve for x, it so happens that all the x's subtract out, and a TRUE statement results.  For example, x +3 = 3+x.  If you subtract x from each side, you get 3=3, or maybe 0=0, which is ALWAYS TRUE.  It's true no matter what x is, so this means the solution is all REAL values of x.  This type of equation is called an IDENTITY.


In yur example, 
2(4x+1)-x=7x+2
8x + 2 - x = 7x + 2
7x + 2 = 7x + 2


Subtract 7x from each side, and you get 2=2, which is always true.  The solution is all real values, and this is called an IDENTITY.


R^2 at SCC