Chapter 1: Linear Equations
- Page ID
- 196640
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: Finding Distances and Midpoints
- Once you are able to plot points and draw lines, the next obvious concept would be measuring distances and determining midpoints. In this section, we will discuss how the distance formula was developed from the Pythagorean Theorem and the midpoint formula is finding the average between the x- and y-values.
- 1.2: Solving Linear Equations
- Solving linear equations is a lifetime skill. Have you ever wondered if you have enough money to get to the end of the month, pay a bill, save enough to buy something, or if you have enough time to reach your destination when driving at a given speed? These are all linear equations. You should be able to solve
- 1.3: Solving Linear Inequalities, Compound Inequalities, and Applications
- Inequalities are everywhere in the real world. For example, you need to spend at least \($75\) to get free shipping or your credit card has a limit of at most \($4000\). These concepts are not difficult to understand. We will study compound inequalities and applications.
- 1.4: Solving Absolute Value Equations and Inequalities
- Absolute value can be thought of like distance. We would never consider driving in reverse to be a negative distance. You are still traveling, but in the opposite direction. That is how we define absolute value, the distance from zero without considering direction.
- 1.5: Functions and Function Notation
- A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships using the correct language and notation.
- 1.6: Domain and Range
- In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions.
- 1.7: Slope of a Line and Graphing Lines
- Slope is one of the most powerful concepts you can understand. For simplicity, we refer to it as rise over run, but the concept that slope represents a rate of change is where the real application comes into play. You deal with slopes every day in your real life when you think about miles per hour, cost per bushel, or bushels to the acre. All of these, and many more, could be thought of as a slope. Go ahead and think about how you use slope in your life, you may be surprised that you use alg
- 1.8: Slope-Intercept, Point-Slope, Standard form of lines, Vertical, Horizontal, Parallel, Perpendicular Lines and Applications
- We introduced slope and graphing lines by plotting points, using the intercepts, or using slope and a point (most generally the \(y\)-intercept). Now we need to be able to apply the slope to real-world rates of change and write the equations of lines so we can apply these concepts to everyday situations.
- 1.9: Solve Systems of Linear Equations with Two Variables
- Often, when solving a real-world problem, there are multiple ideas to consider. Solving a system of two linear equations will ultimately allow us to solve problems involving cost and profit, mixing chemicals, uniform motion and so much more.
- 1.10: Solve Applications with Systems of Equations
- In the previous section, we learned three different methods to solve systems of equations. Now we need to look at some reasons why we may need to solve a system of two or more equations. This is a small subsection of applications. You should also see this topic in the sciences and in business.
- 1.11: Graphing Systems of Linear Inequalities
- When we solve a system of equations, we find one solution that satisfies both of the given equations. The point of intersection is still very important but now we must consider a range of values. We are looking for a region that satisfies both inequalities.
- 1.12: Solve Systems of Equations Using Matrices
- A matrix is a way to store data and then easily manipulate the data without the variables. You can take classes dealing with matrices and matrix theory. This section will introduce you to matrices and show you how to use technology to solve a system of equations.