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

how do you get the 2/50

Abba Reply

number of sport play by 50 student construct discrete data

Aminu Reply

width of the frangebany leaves on how to write a introduction

Theresa Reply

Solve the mean of variance

Veronica Reply

Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.

kenneth

what is error

Yakuba Reply

Is mistake done to something

Vutshila

anas

What is the life teble

anas

Jibrin

statistics is the analyzing of data

Tajudeen Reply

what is statics?

Zelalem Reply

how do you calculate mean

Gloria Reply

diveving the sum if all values

Shaynaynay

let A1,A2 and A3 events be independent,show that (A1)^c, (A2)^c and (A3)^c are independent?

Fisaye Reply

what is statistics

Akhisani Reply

data collected all over the world

Shaynaynay

construct a less than and more than table

Imad Reply

The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.

Aschalew Reply

Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?

Akshay Reply

Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits

Akshay

Bright

Sorry i want to learn more about this question

Bright

Someone help

Bright

a= 0.20233 b=0.3384

Sufiyan

Shaynaynay

How do I interpret level of significance?

Mohd Reply

It depends on your business problem or in Machine Learning you could use ROC- AUC cruve to decide the threshold value

Shivam

how skewness and kurtosis are used in statistics

Owen Reply

yes what is it

Taneeya

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

©flickr: Urko	Music Reviewer MCQ By Edgar Delgado Start Quiz
©flickr: Donnie	How well do you know Beanie Boos? By Angela Eckman Start Quiz
	3 Arts Society: Theater 3 By Jonathan Long Start Quiz
	NCE Ch 10 Professional Orientation By Anh Dao Start Quiz
©flickr: Abraham	2 Biology 2 By Sarah Warren Start Test
	Concepts of biology By OpenStax Read Online Course
	English Proficiency Test By Anindyo Mukhopadhyay Start Quiz
	15 AP Key Terms 15 Autonomic Nervous System By OpenStax Start Key Terms
	7 Psychology MCQ 2010 Final Exam By John Gabrieli Start Exam
	2 BOD - Neuropathology By Brooke Delaney Start Exam

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] )