If you prefer, 1 + 0 x = 2. We agree that is an equation. It has no solution. Typically in math we don't write 0x in expressions.
Suppose we start with 2x+5 = x - 4 + x. If we solve this by showing all work we'll end up at the end with
5 = -4
At this point students are told this means that there are no solutions. But what they miss is what an equation really means. Anyone who really understands the concept of an equation can easily give an example of an equation with no solution.
An equation is a statement that two expressions are equal. What one tries to find is the solution set. That is, the collection of tuples (if more than one variable is present) that makes the expressions equal when evaluated at the tuple. An equation is really a question. How can these two expressions be evaluated at the same input to produce the same output?
Lastly, We agree that f(x) = 1 and g(x) = 0 are valid functions. Here is the equation 1 = 0 stated in a more sophisticated way. f(x) = g(x) where f and g are defined above.
Suppose we start with 2x+5 = x - 4 + x. If we solve this by showing all work we'll end up at the end with
5 = -4
At this point students are told this means that there are no solutions. But what they miss is what an equation really means. Anyone who really understands the concept of an equation can easily give an example of an equation with no solution.
An equation is a statement that two expressions are equal. What one tries to find is the solution set. That is, the collection of tuples (if more than one variable is present) that makes the expressions equal when evaluated at the tuple. An equation is really a question. How can these two expressions be evaluated at the same input to produce the same output?
Lastly, We agree that f(x) = 1 and g(x) = 0 are valid functions. Here is the equation 1 = 0 stated in a more sophisticated way. f(x) = g(x) where f and g are defined above.