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

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

hi guys good evening to all

Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

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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:

Get Jobilize Job Search Mobile App in your pocket Now!

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