7.9: Summary of Key Concepts
- Page ID
- 60040
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Summary of Key Concepts
Graph of a Function
The geometric representation (picture) of the solutions to an equation is called the graph of the equation.
Axis
An axis is the most basic structure of a graph. In mathematics, the number line is used as an axis.
Number of Variables and the Number of Axes
| An equation in one variable requires one axis | One-dimension |
| An equation in two variables requires two axes. | Two-dimensions |
| An equation in three variables requires three axes. | Three-dimensions. |
| An equation in \(n\) variables requires \(n\) axes | \(n\)-dimensions. |
Coordinate System
A system of axes that is constructed for graphing an equation is called a coordinate system.
Graphing an Equation
The phrase graphing an equation is interpreted as meaning geometrically locating the solutions to that equation.
Uses of a Graph
A graph may reveal information that may not be evident from the equation.
Rectangular Coordinate System \(xy\)-Plane
A rectangular coordinate system is constructed by placing two number lines at 90∘ angles. These lines form a plane that is referred to as the \(xy\)-plane.
Ordered Pairs and Points
For each ordered pair \((a,b)\), there exists a unique point in the plane, and for each point in the plane we can associate a unique ordered pair \((a,b)\) of real numbers.
Graphs of Linear Equations
When graphed, a linear equation produces a straight line.
General Form of a Linear Equation in Two Variables
The general form of a linear equation in two variables is \(ax+by=c\), where \(a\) and \(b\) are not both \(0\).
Graphs, Ordered Pairs, Solutions, and Lines
The graphing of all ordered pairs that solve a linear equation in two variables produces a straight line.
The graph of a linear equation in two variables is a straight line.
If an ordered pair is a solution to a linear equation in two variables, then it lies on the graph of the equation.
Any point (ordered pair) that lies on the graph of a linear equation in two variables is a solution to that equation.
Intercept
An intercept is a point where a line intercepts a coordinate axis.
Intercept Method
The intercept method is a method of graphing a linear equation in two variables by finding the intercepts, that is, by finding the points where the line crosses the \(x\)-axis and the \(y\)-axis.
Slanted, Vertical, and Horizontal Lines
An equation in which both variables appear will graph as a slanted line.
A linear equation in which only one variable appears will graph as either a vertical or horizontal line.
\(x=a\) graphs as a vertical line passing through \(a\) on the \(x\)-axis.
\(y=b\) graphs as a horizontal line passing through \(b\) on the \(y\)-axis.
Slope of a Line
The slope of a line is a measure of the line’s steepness. If \((x_1,y_1)\) and \((x_2,y_2)\) are any two points on a line, the slope of the line passing through these points can be found using the slope formula.
\(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{\text{ vertical change }}{\text{ horizontal change }}\)
Slope and Rise and Decline
Moving left to right, lines with positive slope rise, and lines with negative slope decline.
Graphing an Equation Given in Slope-Intercept Form
An equation written in slope-intercept form can be graphed by
- Plotting the \(y\)-intercept \((0,b)\).
- Determining another point using the slope, \(m\).
- Drawing a line through these two points.
Forms of Equations of Lines
General Form: \(ax + by + c\)
Slope-Intercept Form: \(y = mx + b\)
To use this form, the slope and \(y\)-intercept are needed
Point-Slope Form: \(y - y_1 = m(x - x_1)\)
To use this form, the slope and one points, or two points, are needed.
Half-Planes and Boundary Lines
A straight line drawn through the plane divides the plane into two half-planes. The straight line is called a boundary line.
Solution to an Inequality in Two Variables
A solution to an inequality in two variables is a pair of values that produce a true statement when substituted into the inequality.
Location of Solutions to Inequalities in Two Variables
All solutions to a linear inequality in two variables are located in one, and only one, half-plane.