# Exploring the Nature of Solutions

Solve the following equations.  What do you notice/wonder about the solutions?

1. $$2x-3+ 1 = 2(x-1)$$
2. $$5(x-1) = 3(x+2)$$
3. $$4(x-2)+2 = x+3(x-2)$$

Solutions:  One solution, no solution, and infinitely many solutions respectively.

Notes: A linear equation can have one solution, no solution, or an infinite number of solutions.

• An equation of the form $$x+a = x+a$$ has infinite solutions since you can use any value for $$x$$ and this statement will always be true.
• An equation of the form $$x+a = x-a$$ has no solutions since there is no value for $$x$$ that will make this statement true.
• An equation of the form $$x+a = b$$ has one solution since there is only one value for $$x$$ that will make this statement true.