Chapter 1
Linear Equations and Functions

1. 1.1 Solving Linear Equations
   1. 1. Example 1.1.1 Solving a Linear Equation
      2. Example 1.1.2 Solving for y
      3. Example 1.1.3 Simple Interest
      4. Example 1.1.4 Investment
      5. Example 1.1.5 Gasoline Prices
2. 1.2 Linear Equations and Their Graphs
   1. 1. Example 1.2.1 Ordered Pair Solutions
      2. Example 1.2.2 Intercepts and Graph of a Line
      3. Example 1.2.3 Intercepts of a Demand Function
      4. Example 1.2.4 Slope-Intercept Form
      5. Example 1.2.5 Point-Slope Form
      6. Example 1.2.6 Temperature Conversion
      7. Example 1.2.7 Salvage Value
      8. Example 1.2.8 Parallel or Perpendicular Lines
3. 1.3 Factoring and the Quadratic Formula
   1. 1. Example 1.3.1 Finding the GCF
      2. Example 1.3.2 Sum and Difference of Squares
      3. Example 1.3.3 Sum and Difference of Cubes
      4. Example 1.3.4 Factoring Trinomials
      5. Example 1.3.5 Factoring Trinomials (revisited)
      6. Example 1.3.6 Factoring by Grouping
   2. The Quadratic Formula
      1. Example 1.3.7 The Quadratic Formula
      2. Example 1.3.8 Zeros of Quadratics
      3. Example 1.3.9 A Quadratic Supply Function
4. 1.4 Functions and their Graphs
   1. 1. Example 1.4.1 Interval Notation
   2. Functions
      1. Example 1.4.2 Finding Domains
      2. Example 1.4.3 Function Values
      3. Example 1.4.4 Function Notation and Piecewise Intervals
      4. Example 1.4.5 Determining Functions
   3. Graphs of Functions
      1. Example 1.4.6 Graph of a Parabola
      2. Example 1.4.7 A Piecewise (segmented) Graph
   4. The Algebra of Functions
      1. Example 1.4.8 Algebra of Functions
      2. Example 1.4.9 Composite Functions
5. 1.5 Laws of Exponents
   1. 1. Example 1.5.1 Using Exponent Laws
      2. Example 1.5.2 Using Exponent Laws (revisited)
      3. Example 1.5.3 Using Fractional Exponents
6. 1.6 Slopes and Relative Change
   1. 1. Example 1.6.1 Another Difference Quotient
      2. Example 1.6.2 Difference Quotients
7. Historical Notes

1.1 Solving Linear Equations

Mathematical descriptions, often as algebraic expressions, usually consist of alphanumeric characters and special symbols.

The name "algebra" has fascinating origins in early Arabic language (Historical Notes).

For example, physicists describe the distance, s, that an object falls under gravity in a time, t, by . Here, the letters s and t represent variables since their values may change while, g, the acceleration of gravity, is considered as constant. While any letters can represent variables, typically, later letters of the alphabet are customary. Use of x and y is generic. Sometimes, it is convenient to use a letter that is descriptive of a variable, as t for time.

Earlier letters of the alphabet are customary for fixed values or constants. However, exceptions are common. The equal sign, a special symbol, is used to form an equation. An equation equates algebraic expressions. Numerical values for variables that preserve equality are called solutions to the equations.

For example, is an equation in a single variable, x. It is a conditional equation since it is only true when . Equations that hold for all values of the variable are called identities. For example, is an identity. By solving an equation, values of the variables that satisfy the equation are determined.

An equation in which only the first powers of variables appear is a linear equation. Every linear equation in a single variable can be solved using some or all of these properties:

1. Substitution - Substituting one expression for an equivalent one does not alter the original equation. For example, is equivalent to or .
2. Addition - Adding (or subtracting) a quantity to each side of an equation leaves it unchanged. For example, is equivalent to or .
3. Multiplication - Multiplying (or dividing) each side of an equation by a non-zero quantity leaves it unchanged. For example, is equivalent to or .

Here are examples of linear equations: , , . They are linear in one, two, or three variables, respectively. It is the unit exponent on the variables that identifies them as linear.

By "solving an equation" we generally intend the numerical values of its variables.

To Solve Single Variable Linear Equations

1. Resolve fractions.
2. Remove grouping symbols.
3. Use addition (and/or subtraction) to have variable terms on one side of the equation.
4. Divide the equation by the variable's coefficient.
5. As a check, verify the solution in the original equation.

Example 1.1.1 Solving a Linear Equation

Solve .

Solution:

To remove fractions, multiply both sides of the equation by 6, the least common denominator of 2 and 3. (Step 1 above)

The revised equation becomes

Next, remove grouping symbols (Step 2). That leaves

Now, subtract 4x and add 48 to both sides (Step 3). Now,

Finally, divide both sides by the coefficient 5 (Step 4). One obtains .

The result, , is checked by substitution in the original equation (Step 5):

Equations often have more than one variable. To solve linear equations in several variables simply bring a variable of interest to one side. Proceed as for a single variable regarding the other variables as constants for the moment.

If y is the variable of interest in , it can be written as regarding x and z as constants for now.

Example 1.1.2 Solving for y

Solve for y: .

Solution:

Move terms with y to one side of the equation and any remaining terms to the opposite side. Here, . Next, divide both sides by 4 to yield .

Example 1.1.3 Simple Interest

"Interest equals Principal times Rate times Time" expresses the well-known Simple Interest Formula, . Solve for the time, T.

Solution:

Grouping, so PR becomes a coefficient of T. Dividing by PR gives .

Mathematics is often called "the language of science" or "the universal language". To study phenomena or situations of interest, mathematical expressions and equations are used to create mathematical models. Extracting information from the mathematical model provides solutions and insights. Mathematical modeling ideas appear throughout the text. These suggestions may aid your modeling skills.

To Solve Word Problems

1. Read problems carefully.
2. Identify the quantity of interest (and possibly useful formulas).
3. A diagram may be helpful.
4. Assign symbols to variables and other unknown quantities.
5. Use symbols as variables and unknowns to translate words into an equation(s).
6. Solve for the quantity of interest.
7. Check your solution and whether you have answered the proper question.

Example 1.1.4 Investment

Ms. Brown invests $5000 at 6% annual interest. Model her resulting capital for one year.

Solution:

Here the principal (original investment) is $5000. The interest rate is 0.06 (expressed as a decimal) and the time is 1 year.

Using the simple interest formula, , Ms. Brown's interest is

After one year a model for her capital is .

Example 1.1.5 Gasoline Prices

Recently East Coast regular grade gasoline was priced about $3.50 per gallon. West Coast prices were about $0.50/gallon higher.

1. What was the average regular grade gasoline price on the East Coast for 10 gallons?
2. What was the average regular grade gasoline price on the West Coast for 15 gallons?

Solution:

1. On average, a model for the East Coast cost of ten gallons was .
2. On average, a model for the West Coast of fifteen gallons was .

Consumption as a function of disposable income can be expressed by the linear relation , where C is consumption (in $); x, disposable income (in $); m, marginal propensity to consume and b, a scaling constant. This consumption model arose in Keynesian economic studies popular during The Great Depression of the 1930s.

Exercises 1.1

In Exercises 1-6 identify equations as an identity, a conditional equation, or a contradiction.

1. In Exercises 7-27 solve the...