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

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.

John Reply

taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.

Faith

While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma

Kamara

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Ummi Reply

omeprazole

Kamara

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Renee

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Renee

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Kamara

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Kamara

Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit

Patrick

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Nency Reply

Not really sure

Eli

to drain extracellular fluid all over the body.

asegid

The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of

asegid

to transport fluids fats proteins and lymphocytes to the blood stream as lymph

Adama

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Oyindarmola Reply

Anatomy is the identification and description of the structures of living things

Kamara

what's the difference between anatomy and physiology

Oyerinde Reply

Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.

AI-Robot

what is enzymes all about?

Mohammed Reply

Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems

Kamara

yes

Prince

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Kamara

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function of digestive

Ali

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Diya Reply

37 degrees selcius

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37°c

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Mark

36.5

Simon

37°c

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the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature

Stephanie

37A c

Wulku

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Diya Reply

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Diya Reply

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acid

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Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)

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

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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:

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