Does an equation always have 1 solution?

Answer 1

No, not at all!

Here are a few examples. I try to keep them simple, to get the point across, but more complicated examples are also possible.

x = 5: If x = 5, the equation is true, else it is false. Exactly one solution.

x = x + 1: Whatever value you put for "x", the right side will be one more than the left side. Zero solutions.

(x-1)(x-2) = 0: When any of the factors is zero, the product is zero. Therefore, x can be 1 or 2. Two solutions.

x(x+2)(x-5)(x-10) = 0: Similar as the previous example, but x can be 0, -2, 5, 10. Four solutions.

x = x: Any value you put for "x", the equation will be true. Infinite solutions - the entire set of real numbers.

sin x = 0: The sine is a periodic function; it has the value 0 for x = 0, pi, 2 x pi, 3 x pi, etc. (this is in radians; equivalent to 0 degrees; 180 degrees; 360 degrees; etc.). Infinite solutions, but only specific points.