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7.9: Summary of Key Concepts

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    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.


    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.


    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

    1. Plotting the \(y\)-intercept \((0,b)\).
    2. Determining another point using the slope, \(m\).
    3. 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.

    This page titled 7.9: Summary of Key Concepts is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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