2.1 - Linear Equations and Modeling

Definitions

Equation
A statement that two expressions are equal
Solutions
Values which make the equation true
Identity
An equation which is true for every real number in the domain
Contradiction
An equation which is false for every real number in the domain
Conditional equation
An equation which may be true or false depending on the values of the variables.
Equivalent equations
Equations having the same solution set.
Linear equation in one variable
Equation that can be written as ax+b=0, where a and b are reals and a doesn't equal zero. If a did equal zero, it would be a constant equation and an identity if b=0 or a contradiction if b≠0.
Extraneous solutions
Solutions which satisfy an "equivalent" equation, but not the original equation. They can be introduced by multiplying or dividing by an expression containing a variable. They can also be introduced by applying a non-one-to-one function to both sides (like squaring both sides). You should always check your answer when there is a possibility that you have introduced an extraneous solution.
Mathematical model
An algebraic equation used to solve a problem that occurs in real life.

Equivalent Operations

The following operations can be used to generate an equivalent equation

Solving Equations

These are general guidelines, individual problems may vary.

Linear Equations

  1. Eliminate any grouping symbols
  2. Use addition/subtraction to move all terms containing the variable to one side and all terms without the variable to the other side. Simplify. If there is more than one occurrence of the variable, you may need to factor the variable out.
  3. Use multiplication/division to get the variable by itself.
  4. Check your answer

Decimals

  1. Multiply every term of both sides of the equation by a power of 10 (10, 100, 1000, etc) to eliminate the decimals. Note that expressions like 0.42 (x + 2) are just one term because of the multiplication. Either multiply the 0.42 by 100 (preferred), or the (x+2) by 100 (ok, but doesn't get rid of the decimals), but not the 0.42 by 100 and the (x+2) by 100 (you've really multiplied by 10,000 if you do that).
  2. Finish the problem like it was a regular linear equation.

Note that there is no requirement to eliminate the decimals. Some people prefer to work with the decimals. Other people find it more difficult. It is a personal preference, but most people will eliminate the decimals.

Fractions

Fractions are your friends. They will hang out with you on the weekends when no one else will. However, most people find it difficult to work with fractions, and although they are your friends, solving equations becomes much easier if they're not around.

  1. Find the least common denominator out of all of the denominators. Be sure to factor any denominator first if possible.
  2. Note somewhere that the values which make the LCD zero can not be used. The LCD is a collection of all things in the denominator, and so anything that makes it zero would cause division by zero, and this is a very bad thing. Our goal is to get rid of the fractions, so in the next step there will be no more denominator. That means that what was in the implied domain is no longer in the implied domain and must be stated overtly.
  3. Multiply each term of both sides of the equation by the least common denominator. All of the denominators will inverse out and become 1 (and hence don't need to be written).

Note that if you're working with an equation, the fractions can be eliminated. No such claim can be made about working with expressions, however. It is the multiplicative property of equality which allows us to eliminate the denominators, and you just don't have equality when you're working with expressions.

Mathematical Models

Mathematical modeling is the process of taking a verbal description of a problem, assigning labels to the unknown quantities and forming a mathematical model or algebraic equation.

Keywords

Here are some words and what they usually mean.

Equality
is, are, will be, represents
Addition
sum, plus, greater, increased by, more than, exceeds, total
Subtraction
difference, minus, less, decreased by, subtracted from, reduced by, the remainder
Multiplication
product, multiplied by, twice, times, percent of
Division
quotient, divided by, ratio, per

Formulas

There are common formulas, formulas from geometry, and conversions on the back cover of your text.

You can ignore the formulas dealing with the trigonometric functions (sin, cos, tan) and the area of a sector. You don't need to memorize, but don't forget that they're there, the formulas for an equilateral triangle, circular ring, right circular cone, and frustrum.

Definitely know (be able to recall and apply) the formulas for the area, perimeter/circumference of a triangle, parallelogram (includes rectangles and squares), and circle. Also be aware that other shapes, like a trapezoid, can be broken down into simpler shapes (like two triangles with the same height).