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The Guts of Algebra

The Solving Equations Series

A Guts of Math Team Guide

Copyright 2014 by the Guts of Math Team

All rights reserved.

http://www.gutsofmath.wordpress.com

Contents

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Foreword

This is a no-nonsense guide that presents exactly what the title indicates: the guts or most fundamental principles involved in solving linear, quadratic, exponential, and logarithmic equations in algebra. If you are interested in a quick review of the material or a concise, efficient introduction to the topic, this e-book set is for you.

The Guts of Math Team strongly urges you to read and digest the brief summary before proceeding to the worked-out problems. Once you have read through these problems, try them again on your own. You should then be ready to test yourself with the practice problems; the solutions are written up for you at the end in a separate section. Great care has been taken to spell out the approach employed and all the steps taken.

The only way to get better at mathematics is to do it, so please do not just read this material passively!

We wish you much success in your studies!

The Guts of Math Team

Linear Equations

Previous Knowledge Needed

Basic Arithmetic: How to multiply, divide, add, subtract real numbers; how to simplify fractions.

Algebra of Like Terms : How to combine terms involving the same variables. For example,

whereas

and the latter is as simplified as possible.

Both Sides Rule : What you do to one side of an equation (multiply/divide by a number, add/subtract a number), you must do to the other.

Distributive Law for Multiplication : Given real numbers , , and , we have

Distributivity also works on the other side:

Commutative Law for Addition : Given real numbers and , we have

Summary

A linear equation in the variable is an equation that can be written in the form:

where and are constants. Usually, , , and are taken to be real numbers, and we will make this assumption in this guide. Notice that the definition given here uses the phrase can be written in the form, which suggests that an equation may be linear but may not already be in that form. For example,

is not in the form , but it can be put into this form by applying some algebra (see Problem 3).

Once a linear equation is written in the form , where, ,, and are real numbers, there are three cases.

Case 1 : If , then we simply divide both sides of the equation by to get

or, using the simplification

we find that the solution is

Case 2 : If

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