1: Linear Equations and Lines

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Toward the end of the twentieth century, the values of stocks of internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. The Standard and Poor’s Index tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.

Figure of a bull and a graph of market prices. — Figure $\PageIndex{1}$: Figure of a Bull and a graph of market prices. Standard and Poor’s Index with dividends reinvested (credit "bull": modification of work by Prayitno Hadinata; credit "graph": modification of work by MeasuringWorth)

Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties.

Contributors and Attributions

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.

Learning Objectives

In this chapter, you will learn to:

Solve linear equations
Solve linear inequalities, expressing solutions on a number line and in interval notation
Graph linear equations
Find the equation of a line
Apply linear models to data

1.1: Solving Linear Equations and Inequalities
An equation is a statement indicating that two algebraic expressions are equal. A linear equation with one variable, x , is an equation that can be written in the standard form ax+b=0 where a and b are real numbers and a≠0 . A solution to a linear equation is any value that can replace the variable to produce a true statement.
- 1.1E: Exercises - Solving Linear Equations and Inequalities
1.2: Graphing Linear Equations
Equations whose graphs are straight lines are called linear equations. A line is completely determined by two points. Therefore, to graph a linear equation we need to find the coordinates of two points. This can be accomplished by choosing an arbitrary value for x or y and then solving for the other variable.
- 1.2E: Exercises - Graphing Linear Equations
1.3: Determining the Equation of a Line
- 1.3E: Exercises - Determining the Equation of a Line
1.4: Linear Applications
Now that we have learned to determine equations of lines, we get to apply these ideas in a variety of real-life situations.
- 1.4E: Exercises - Linear Applications
1.5: Fitting Linear Models to Data
Scatter plots show the relationship between two sets of data. Scatter plots may represent linear or non-linear models. The line of best fit may be estimated or calculated, using a calculator or statistical software. Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. The correlation coefficient, r , indicates the degree of linear relationship between data.
- 1.5E: Exercises - Fitting Linear Models to Data
1.6: Chapter 1 Review