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**Number Of Solutions To An Equation**: At a Glance

- Topics At a Glance
**Solutions to Equations**- Checking Solutions to Equations
**Number of Solutions to an Equation**- Equivalent Equations
- Solving Equations with One Variable
- Adding and Subtracting Constants
- Checking Answers
- Adding and Subtracting Variables
- Multiplication and Division
- Complicated Equations
- Simplifying Equations
- Eliminating Fractions
- Keeping Both Solutions
- When You Get Stuck
- Solving Equations with Multiple Variables
- Solving Equations for Expressions
- Keeping Answers Pretty
- Factoring
- Geometry
- Single-Variable Inequalities
- Strict Inequalities
- Equivalent Inequalities
- Inequalities that Allow Equality
- Solving Inequalities
- In the Real World
- Fitting Things in Spaces
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Okay, algebristas,* different equations can have different numbers of solutions. Right now, we are only counting solutions from the real numbers. Some equations have solutions that are imaginary numbers, but we'll get to those later. No, you don't need to send us a Thank You card. Your words are enough.

*You know, people who serve mathuccinos. Alternatively, you.

- The equation
*x*= 5 has only one solution: the number 5. - The equation
*z*^{2}= 4 has two solutions:*z*= 2 and*z*= - 2. - The equation
*x*=*x*has infinitely many solutions: any value of*x*will work, since*x*is always equal to itself. - The equation
*y*^{2}= - 5 has no real number solutions because the square of any real number is positive.

We interrupt this program to bring you a History Snack. Don't let it ruin your appetite.

Diophantus was a famous mathematician who's sometimes called "The Father of Algebra." Although with the way he got around, who *wasn't* he the father of? #ancientgossip

Anyway, D-man only liked positive rational solutions to equations. He would probably call a lot of the equations we will be solving "absurd" since they have negative solutions. It's hard to blame him, thoughâ€”after all, there's no such thing as a negative child. (He wishes. Oh, burn.)

Exercise 1

Figure out how many solutions there are to the equation *x* = *x* + 1.

Exercise 2

Figure out how many solutions there are to the equation *w* + 3 = 7.

Exercise 3

Figure out how many solutions there are to the equation 2*x* = *x*.

Exercise 4

Figure out how many solutions there are to the equation *x*^{4} = 1.