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**Adding And Subtracting Variables**: 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

Just as we can add and subtract constants from both sides of an equation, we can also add and subtract copies of the variable from both sides of the equation. Therefore, if the same variable appears on both sides of the equation, we can reduce them as much as possible in order to get one variable on one side of the equation. It's always nice to have just a single "*x*" (especially when following a treasure map, you know, as you do). We need to add or subtract the same number of copies of the variable from each side.

Remember that our mission, if we choose to accept it, is to get the variable on one side of the = sign and a number on the other side.

Solve the equation 4*x* = 5*x* + 1. Check your answer.

We'd like to have all the *x*'s by themselves on one side of the equation, so we subtract 4 copies of *x* from each side to find that

0 = *x* + 1

Yay—so few copies! This will shave a bundle off our Kinko's bill.

We know what to do from here: subtract 1 from each side of the equation, and write -1 = *x*.

To check our answer, we evaluate the left side of the original equation and the right side of the original equation individually for *x *= -1. The left side of the equation, evaluated at *x* = -1, is

4(-1) = -4

The right side of the equation evaluated at *x* = -1 is

5(-1) + 1 = -5 + 1 = -4

Because the two sides of the equation agree when evaluated at *x* = -1, the solution to the equation is indeed *x* = -1. There's one of those negative solutions again. Sorry, Diophantus.

Example 1

Solve the equation - |

Exercise 1

Solve the equation 5*x* - 10 = 6*x* + 3. Check your answer.

Exercise 2

Solve the equation -*y* + 9 = 14. Check your answer.

Exercise 3

Solve the equation -8*z* + 20 = -7*z* + 5. Check your answer.