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Chapter 6—Expressions and Equations

6-7 Solving Equations

 When you complete the work for this section, you should be able to: Give examples of true, false, and conditional equations. Describe what it means to solve an equation. Solve equations of the form x + a = b Solve equations of the form ax = b

Recall that an equation is a statement of equality between two expressions.

Purely Numerical Equations

When there are no variables in an equation, the equation is either true or false.

 True Equations False Equations 4 = 4 3 = 4 3 + 2 = 5 5 + 3 = 2 82 = 8 x 10 + 2 10 = 22 + 3

 Note This thing about 4 = 4 being true and 3 = 4 being false might seem so obvious that it's silly. Well, most of the principles of mathematics are simple ... really!! You just have to learn to see such things from their simpler points of view.

Conditional Equations

Equations may be true, false, or conditional. You have already seen that purely numerical equations can be true or false. Include a variable, however, and you get a conditional equation.

0 = 0 is definitely true
0 = 2 is definitely false
2x = 8 is conditional. It can be true or false, depending on the value assigned to x.
 Definition A conditional equation is one that can be true or false, depending on the values assigned to its variable(s).

Once a numerical value is assigned to the variables in a conditional equation, it becomes either true or false.

Example

3 + x = 5 is a conditional equation.

If you set x equal to 1, it becomes a false equation:

3 + 1 = 5
4 = 5        Definitely false

If you set x equal to 2, it becomes a true equation:

3 + 2 = 5
5 = 5        Definitely true

Solving Some Basic Equations

 Definition The process of finding values for variables that produce a true equation is called solving the equation.

 Note Most of the work in algebra and higher math involves solving equations.  So you should strive to master the basic ideas now.

Consider this conditional equation: x + 1 = 5.

This says: "Adding 1 to some number x is equal to 5."
In order to change this into a true equation, you need to determine the correct value for x.
Well, if you think for a bit, you can see that x must be equal to 4. Or, in other words, 4 + 1 = 5.
The conditional equation becomes a true equation when x = 4.
We also say that x = 4 is the solution to the equation, x + 1 = 5.

Also, you can check your solution by substituting the solution into the original equation:

In this example, you believe x = 4. Evaluating the original equation: 4 + 1 = 5. And that is a true equation.

Here are a few more examples of solving equations and checking the results:

Solve         x + 8 = 15
Solution:    x = 7
Check:        7 +  8  = 15

Examples:

Examples & Exercises

 Use these interactive Examples & Exercises to strengthen your understanding and build your skills. Continue working the problems until you can do them consistently without error.: