# Question #ac581

Nov 20, 2016

No, not every equation has an answer..
There is a whole range of possibilities.

#### Explanation:

No, not every equation has an answer.

A linear equation usually has one answer ,

while a quadratic $\left({x}^{2}\right)$ usually has two answers

and a cubic equation (${x}^{3}$ usually has three answers and so on.

There are some equations (which are actually identities) which solve to give the result:
$0 = 0$

This is a true statement, but there is no variable left to solve for.
This means that t he variable can have any value. ..
Thus there are infinitely many solutions/answers.

On the other hand, sometimes equations solve to give:

$0 = 9$ (or any other value)

This is a false statement without a variable. It means that equation cannot be solved, there is no solution, and the variable has no value.

In solving an equation you might end up with $\frac{x}{0}$.
Division by 0 is not allowed, so this solution would be undefined.

Some equations end up with the result.

$x = \sqrt{-} 16$ (or any other negative number)

These equations do not have any real number as a solution and you end up working in the world of the complex numbers .

Nov 20, 2016

Not every equation has a solution. Every equation has a solution set. But the solution set for some equations is empty. (There are no solutions.)

#### Explanation:

Sometimes we give solutions of equations using a list, other times we use the solution set. (Don't be intimidated by 'set'. A set is a container -- like a box-- that may or may not have things in it.)

Examples

For $x + 3 = 9$ the solution is $6$ and the solution set is $\left\{6\right\}$

For $2 x + 5 = x + 7$ the solution is $2$ and the solution set is $\left\{2\right\}$

For ${x}^{2} = 9$ the solutions are $- 3$ and $3$ and the solution set is $\left\{- 3 , 3\right\}$

For $x + 1 = x$ there are no solutions. We sometimes say the solution does not exist.
There is still a set (box) of solutions, there just isn't anything in the box (set). The solution set is $\left\{\right\}$ which we usually denote by a circle with a slash through it. Like this: $\cancel{O}$. We call it the empty set.

For $2 \left(x + 3\right) = 2 x + 6$, any number (every number), when substituted for $x$ will make the statement true, so the list of solution has all numbers on it. We can't finish writing that. We say one of: "every number is a solution", or "the solutions are all numbers" (sometimes we specify "real numbers"), For the solution set we use the biggest set we are working with. If we're working in the real numbers we say: the solution set is (the set of) all real numbers (or 'all reals'). We use the notation $\mathbb{R}$ for the set of all real numbers.