7.8 Summary of key concepts

Elementary algebra Page 1 / 1

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benefits students who take future courses that use graphs to display information.The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines. This module presents a summary of the key concepts of the chapter "Graphing Linear Equations and Inequalities in One and Two Variables".

Summary of key concepts

Graph of a function ( [link] )

The geometric representation (picture) of the solutions to an equation is called the graph of the equation.

Axis ( [link] )

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 ( [link] )

\begin{array}{l} An ​ equation in one variable requires one axis . ​ & One-dimension . \\ An equation in two variable requires two axes . & Two-dimensions . \\ An equation in three variable requires three axes . & Three-dimensions . \\ \cdot \\ \cdot \\ \cdot \\ An equation in n variable requires n axes . & n -dimensions . \end{array}

Coordinate system ( [link] )

A system of axes that is constructed for graphing an equation is called a coordinate system .

Graphing an equation ( [link] )

The phrase graphing an equation is interpreted as meaning geometrically locating the solutions to that equation.

Uses of a graph ( [link] )

A graph may reveal information that may not be evident from the equation.

Rectangular coordinate system $x y$ -plane ( [link] )

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

x y

-plane.

Ordered pairs and points ( [link] )

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 ( [link] )

When graphed, a linear equation produces a straight line.

General form of a linear equation in two variables ( [link] )

The general form of a linear equation in two variables is

a x + b y = c,

where

a

and

b

are not both 0.

Graphs, ordered pairs, solutions, and lines ( [link] )

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 ( [link] )

An intercept is a point where a line intercepts a coordinate axis.

Intercept method ( [link] )

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 ( [link] )

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 ( [link] )

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 = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{vertical change}{horizontal change}

Slope and rise and decline ( [link] )

Moving left to right, lines with positive slope rise, and lines with negative slope decline.

Graphing an equation given in slope-intercept form ( [link] )

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 ( [link] )

\begin{matrix} \underline{General form} & \underline{Slope-intercept form} & \underline{point-slope from} \\ a x + b y = c & y = m x + b & y - y_{1} = m (x - x_{1}) \\ \begin{array}{l} To use this form, the \\ slope and y -intercept \\ are needed . \end{array} & \begin{array}{l} To use this form, the \\ slope and one point, \\ or two points, are needed . \end{array} \end{matrix}

Half-planes and boundary lines ( [link] )

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 ( [link] )

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 ( [link] )

All solutions to a linear inequality in two variables are located in one, and only one, half-plane.

Questions & Answers

what is phylogeny

Odigie Reply

evolutionary history and relationship of an organism or group of organisms

AI-Robot

Deng

what is biology

Hajah Reply

the study of living organisms and their interactions with one another and their environments

AI-Robot

what is biology

Victoria Reply

HOW CAN MAN ORGAN FUNCTION

Alfred Reply

the diagram of the digestive system

Assiatu Reply

allimentary cannel

Ogenrwot

How does twins formed

William Reply

They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together

Oluwatobi

what is genetics

Josephine Reply

Genetics is the study of heredity

Misack

how does twins formed?

Misack

What is manual

Hassan Reply

discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles

Joseph Reply

what is biology

Yousuf Reply

the study of living organisms and their interactions with one another and their environment.

Wine

discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form

Joseph Reply

what is the blood cells

Shaker Reply

list any five characteristics of the blood cells

Shaker

lack electricity and its more savely than electronic microscope because its naturally by using of light

Abdullahi Reply

advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear

Abdullahi

cell theory state that every organisms composed of one or more cell,cell is the basic unit of life

Abdullahi

is like gone fail us

DENG

cells is the basic structure and functions of all living things

Ramadan

What is classification

ISCONT Reply

is organisms that are similar into groups called tara

Yamosa

in what situation (s) would be the use of a scanning electron microscope be ideal and why?

Kenna Reply

A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,

Hilary

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask

	5 Arts Society: Theater 5 By Jonathan Long Start Quiz
	Computer Skills Literacy MCQ By LaToya Trowers Start Quiz
©flickr: Jonathan	Power By Megan Earhart Start Quiz
©flickr: Justin	Music Appreciation Final Practice By Madison Christian Start Exam
	Concepts of biology By OpenStax Read Online Course
	38 Biology 38 The Musculoskeletal System MCQ By OpenStax Start Quiz
	Deciduous Forest By Hope Percle Start Quiz
	Anthropology Culture Personality By Richley Crapo Start Assignment
	Negotiations Conflict Management BUS403 By Charles Jumper Start Quiz
	Vocabulary Week 1-3 By Rachel Woolard Start Quiz

7.8 Summary of key concepts

Summary of key concepts

Graph of a function ( [link] )

Axis ( [link] )

Number of variables and the number of axes ( [link] )

Coordinate system ( [link] )

Graphing an equation ( [link] )

Uses of a graph ( [link] )

Rectangular coordinate system x y -plane ( [link] )

Ordered pairs and points ( [link] )

Graphs of linear equations ( [link] )

General form of a linear equation in two variables ( [link] )

Graphs, ordered pairs, solutions, and lines ( [link] )

Intercept ( [link] )

Intercept method ( [link] )

Slanted, vertical, and horizontal lines ( [link] )

Slope of a line ( [link] )

Slope and rise and decline ( [link] )

Graphing an equation given in slope-intercept form ( [link] )

Forms of equations of lines ( [link] )

Half-planes and boundary lines ( [link] )

Solution to an inequality in two variables ( [link] )

Location of solutions to inequalities in two variables ( [link] )

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Rectangular coordinate system $x y$ -plane ( [link] )