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

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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

	8 Arts Society: Theater 8 By Jonathan Long Start Quiz
	25 AP 25 Urinary System Essay By OpenStax Start Flashcards
	American Politics MCQ By Nicole Bartels Start Quiz
	Gastrointestinal Pathophysiology 2006 By Tamsin Knox Start Exam
	Business Law Ethics By Kevin Moquin Start Quiz
	Basic Of Computer Exam By Naveen Tomar Start Quiz
	12 Sociology 12 Gender, Sex, and Sexuality MCQ By OpenStax Start Quiz
	Are you a 5sos fan? By Rylee Minllic Start Quiz
	Advertising Promotion BUS210 By Melinda Salzer Start Quiz
	Economy Foundations By Robert Murphy Start Test

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] )