- Combining Techniques in Equation Solving
- Recognizing Identities and Contrdictions

Inside Collection (Textbook): Basic Mathematics Review

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules ((Reference)) and ((Reference))). Objectives of this module: be comfortable with combining techniques in equation solving, be able to recognize identities and contradictions.

- Combining Techniques in Equation Solving
- Recognizing Identities and Contrdictions

In Sections (Reference) and (Reference) we worked with techniques that involved the use of addition, subtraction, multiplication, and division to solve equations. We can combine these techniques to solve more complicated equations. To do so, it is helpful to recall that an equation is solved for a particular variable when all other numbers and/or letters have been disassociated from it and it is alone on one side of the equal sign. We will also note that

To associate numbers and letters we use the order of operations.

- Multiply/divide
- Add/subtract

To undo an association between numbers and letters we use the order of operations in reverse.

- Add/subtract
- Multiply/divide

Solve

Solve

Solve

Solve

Solve

Solve

Solve

Sometimes when solving an equation it is necessary to simplify the expressions composing it.

Solve

Solve

Solve and check each equation.

Often the variable we wish to solve for will appear on both sides of the equal sign. We can isolate the variable on either the left or right side of the equation by using the techniques of Sections (Reference) and (Reference).

Solve

Solve

Solve

Solve

Solve

As we noted in Section (Reference), some equations are identities and some are contradictions. As the problems of Sample Set D will suggest,

- If, when solving an equation, all the variables are eliminated and a true statement results, the equation is an identity.

- If, when solving an equation, all the variables are eliminated and a false statement results, the equation is a contradiction.

Solve

The variable has been eliminated and the result is a true statement. The original equation is an *identity.*

Solve

The variable has been eliminated and the result is a false statement. The original equation is a *contradiction.*

Classify each equation as an identity or a contradiction.

identity,

contradiction,

identity,

contradiction,

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction.

contradiction

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables.

Solve

Solve

Solve

Solve

Solve

Solve

Solve

Solve

Solve

Solve

Solve

Solve

Solve for .

*((Reference))* Simplify

*((Reference))* Find the product.

*((Reference))* Find the product.

*((Reference))* Solve the equation

*((Reference))* Solve the equation

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