linear equation.doc

8/14/2019 Linear equation.doc

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Linear equation

Definition

A linear equationis an algebraic equation in which each term is either a constant or the productof a constant and (the first power of) a single variable.

Linear equations can have one or more variables. Linear equations occur with great regularity inapplied mathematics. While they arise quite naturally when modeling many phenomena, they areparticularly useful since many non-linear equations may be reduced to linear equations byassuming that quantities of interest vary to only a small etent from some !bac"ground! state.Linear equations do not include eponents.

Linear equations in two variables

A common form of a linear equation in the two variables xand yis

where mand bdesignate constants (parameters). #he origin of the name !linear! comes from thefact that the set of solutions of such an equation forms a straight line in the plane. $n thisparticular equation, the constant mdetermines the slope or gradient of that line and the constantterm b determines the point at which the line crosses the y-ais, otherwise "nown as the y-intercept.

%ince terms of linear equations cannot contain products of distinct or equal variables, nor anypower (other than &) or other function of a variable, equations involving terms such as xy, x', y&,

and sin(x) are nonlinear.

Equations of the straight line

Linear equations can be rewritten using the laws of elementary algebra into several differentforms. #hese equations are often referred to as the !equations of the straight line.! $n whatfollows,x, y, t, and are variables* other letters representconstants (fied numbers).

General (or standard) form

$n the general (or standard) form the linear equation is written as+

where Aand Bare not both equal to ero. #he equation is usually written so that A , by

convention. #he graph of the equation is astraight line, and every straight line can be represented

by an equation in the above form. $f Ais nonero, then the x-intercept, that is, the x-coordinate of

the point where the graph crosses the x-ais (where, yis ero), is CA. $f Bis nonero, then the y-
http://en.wikipedia.org/wiki/Constant_termhttp://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttp://en.wikipedia.org/wiki/Line_(geometry)http://en.wikipedia.org/wiki/Coordinatehttp://en.wikipedia.org/wiki/Constant_termhttp://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttp://en.wikipedia.org/wiki/Line_(geometry)http://en.wikipedia.org/wiki/Coordinate


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intercept, that is the y-coordinate of the point where the graph crosses the y-ais (where is ero),

is CB, and theslopeof the line is /AB. #he general form is sometimes written as+

where aand bare not both equal to ero. #he two versions can be converted from one to the other

by moving the constant term to the other side of the equal sign.

Slopeintercept form

where mis the slope of the line and bis the y-intercept, which is the y-coordinate of the location

where line crosses the yais. #his can be seen by letting x0 , which immediately givesy0 b. $t

may be helpful to thin" about this in terms of y0 b1 mx* where the line passes through the point

(, b) and etends to the left and right at a slope of m. 2ertical lines, having undefined slope,

cannot be represented by this form.

Pointslope form

where mis the slope of the line and (x&,y&) is any point on the line.

#he point-slope form epresses the fact that the difference in the ycoordinate between two points

on a line (that is, y/ y&) is proportional to the difference in the xcoordinate (that is, x/ x&). #he

proportionality constant is m(the slope of the line).

Twopoint form

where (x&, y&) and (x', y') are two points on the line with x'3 x&. #his is equivalent to the point-

slope form above, where the slope is eplicitly given as (y'/ y&)(x'/ x&).

4ultiplying both sides of this equation by (x'/ x&) yields a form of the line generally referred toas the symmetric form+

!ntercept form

where aand bmust be nonero. #he graph of the equation has x-intercept aand y-intercept b. #he

intercept form is in standard form with AC0 &aand BC0 &b. Lines that pass through the

origin or which are horiontal or vertical violate the nonero condition on aor band cannot be

represented in this form.
http://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Slope


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Linear equations in two variables

#he equation of a straight line is usually written this way+


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Purpose of Linear Equations

A specific algebraic formula we can use to understand the world around us is the linear equation.%ometimes when we want to understand something we have to account for all the variables

involved. When you see a horse galloping around a race trac" and you wonder 5ust how fast sheis running, you can use the linear equation+ s 0 dt (speed is equal to distance divided by time) todiscover how fast she was running.

"istor#

According to the 6ncyclopedia 7ritannica, !#he earliest etant mathematical tetfrom 6gypt is the 8hind papyrus (c. &9: 7.;.). $t and other tets attest to theability of the ancient 6gyptians to solve linear equations in one un"nown.! #he


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>our :-foot cord would stretch an additional &&&.: feet and you would recoilwell above the swirling waters below.

Sports %pplication

What about our horse at the race trac"@ Let?s figure out how fast she ran her lastrace.We "now that her last race was one mile long and she too" three minutes tofinish. %ince we "now that speed 0 distancetime we would merely need to plug inthe "nown variables.

E 0 & mile minutesE 0 & mile.: (this is divided by 9, and will give us our answer in miles perhour)E 0 ' miles per hour.

&inancial %pplications

Hinally, let us loo" at a financial application of a linear equation. (Hor more linearequation, eamples, including a detailed brea"down of this one, see the 8esourcelin" below.)

%omeone invests I', in two bond mutual funds, a 5un" bond fund and agovernment bond fund. #he 5un" bond fund is ris"y and yields &&J interest. #hesafer government bond fund yields only :J. #he total income for the year fromthe two investments was I&,. Kow much did she invest in each fund@

.&& 1 .:(', -- ) 0 &.&& 1 &,- .: 0 &.9 1 & 0 &,.9 0 E 0 :,.I:, is the amount invested in the 5un" bond and I&:, is the amountinvested in the government bond.

#he above eamples show us that the purpose of the linear equation is to describe relationships

among different variables in a variety of practical applications. >ou might find occasions to

apply linear equations in your own life.


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Broader Use of Linear Equations

Linear equations can describe physics, business and biology. %ystems of linear equations modelphenomena with multiple relationships. #hey are a conceptual foundation for calculus-based

theories of slope and are utilied in numeric approimations.

Ph#sics

6quations of the form y 0 m 1 b can trac" movement. #he constant b defines the startingposition, and m describes steady velocity. #he variable is the time during which movement isoccurring at velocity m. #ogether, this information gives the total distance covered--y.

Economics

A fundamental tool of the economist is the supply and demand curve that shows how much of aparticular good a supplier will produce and how much the consumer will purchase at a givenprice. A linear equation produces the relationship between price and units and can therefore be

used to determine the equilibrium price -- the price at which the supplier will produce the eactnumber of units demanded by the consumer. #his is useful for creating economic models, thoughin practice supply and demand do not generally behave in such a well-ordered fashion.

'usiness

$n finance, if you start with I: (b 0 I:) and wor" for 9 hours at I&hr (m 0 I&hr, 0 9 hours),after those 9 hours there is m 1 b 0 &(9) 1 : 0 I9:. Linear equation is widely used in business.

'iolog#

;ric"et chirping frequency correlates to outside temperature. A pro5ect eploring this wasconducted by ane 4. 7rown of the Mniversity Nf $llinois ;ollege Nf 6ducation.

ultiple Linear Equations

%ystems of linear equations can describe several ob5ect types. %cenarios involve coin types whendealing with change and amount and total mass of lightheavy weights at a gym.

'road athematical $elevance

Linear equations are relevant to advanced mathematics. Dewton?s method is based on repeatedapplications of linear equations.

!ntro to alculus

%tudents soon understand that a steeper line has a larger (absolute) value of m. ;alculus-basedconcepts of slope can be introduced by referring to linear equations as a beginning eample.


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Examples of Linear Equations Used in Real Life

#he value of linear equations may be appreciated more when applied to real life eamples.Linear equations model the relationship between two variables and the effect that a change onone variable has on the other.

*emand urve

#he demand curve illustrates the relationship between the price of a product and the quantityconsumers are willing to buy. At lower prices, consumers may be willing to buy more but buyless as prices increase.

Suppl# urve

#he supply curve shows the relationship between prices and quantities of products businesses arewilling to produce. 7usinesses determine how much of their products to sell at a price that

maimies profits.

!nterest $ates

Linear equations may also model the relationship between investment and interest rates whichshow that as interest rates increase the general level of investment will decrease and increase asinterest rates decrease.

&oreign urrencies

>ou can also compare currency echange rates with a linear equation to see how changes in the

value of the Argentine peso affect the M.%. dollar.

%cceleration and *istance Traveled

A linear equation can be used to calculate an increase or decrease in speed after a fied period ofsteady acceleration or deceleration. #he final speed will be the initial speed plus the rate ofacceleration multiplied by the period of time the car accelerated. %imilarly, if a vehicle travels ata fied speed, a linear equation can be used to calculate the distance it travels over a fied period.


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How to Use Linear Equations in Everyday Life

=eople tend not to thin" in terms of equations and formulas in their daily lives. #hey uselanguage to describe the situation. 7ut words can be translated into the language of mathematics.#a"e a very simple eample+ A mother has to divide si apples among three children.

6ffortlessly she reaches the conclusion that each child gets two apples. What she has used is themathematical function of division to reach the answer+ 90'.

4athematics is used in everyday life, although most of the time you won?t even realie you areusing it. 6verything from dividing a ca"e into equal portions to determining the sale price of anitem on the clearance rac" requires mathematics. Nne of the most commonly used mathematicaltools is linear equations. Linear equations permit their users to solve comple and fleibleproblems with a set of simple equations. #hose things that never change in the equation arecalled constants. #he things that do change are variables and are traditionally represented byletters.

Situation +,

;alculate your cab fare for a trip home by forming a linear equation. #he boarding rate that thedriver requires is a constant* the meter rate is also a constant, but must be multiplied by how faryou went. %o, if the meter rate is I', the boarding rate is IG and distance is represented by !!,the linear equation would be ' 1 G 0 cab fare.

Situation -,

;alculate your fuel efficiency. Let?s say your car gets G miles to the gallon when you travelbelow : miles per hour and miles per gallon when you travel above : miles per hour.

Fetermine how much fuel you?ll need to drive somewhere by calculating how often you?ll travelunder and over : miles per hour. %o the equation could be a 1 bG 0 fuel needed.

Situation .,

Fetermine how much weight you?ll lose by creating a calorie-burning equation. Hor instance,you?ll burn &' calories per hour of wal"ing. %o multiply &' calories by the number of hoursyou wal"ed and you?ll determine how many calories you burned. $f you wal"ed less than & hour,use the fraction for how long you did wal", such as one-quarter hour. #he equation for thiswould simply be &' 0 burnt calories.

Situation /,

%uppose your office is miles away from home. >ou have to get there at a.m., and "now thatthe traffic is moving at 9 miles per hour. #o find out the time you should leave home, translatethe word problem into an equation+ time ta"en 0 distance divided by the rate of travel. %o t (time)0 d (distance)r (rate), and t09. %o t0&' or half an hour. #o reach the office at a.m., youshould leave at C+ a.m.


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Situation /,

Kow many minutes are there in four hours@ Let 0 the number of hours, and y 0 the number ofminutes. 7y definition, there are 9 minutes in one hour. %o you can write a linear equation todescribe this relationship+ y 0 9. #he number of minutes equals 9 times the number of hours.

Hor eample, let 0 G. #hen plug the number into the linear equation to get y 0 9OG. %o y 0 'Gminutes.

Kow Are Linear 6quations Msed in 7usiness@

Linear functions have a ma5or application in business decision ma"ing. At this very moment,lowest-cost and higher-profit decisions are being made somewhere with information generatedby computers manipulating linear relationships. Linear functions are of special importance sincethey are easy to manipulate and understand both graphically and symbolically. 7ecause of thismany non-linear situations are analyed using linear approimations. #he intent of theapproimation is to "eep things as simple as possible.

Linear equations are about the use of "nown quantities to discover un"nown quantities. 7usinessis about the echange of money, and any unit of money is measured as a quantity. #he money isechanged for other quantities--for hours of wor", for tons of raw materials or for the volts ofelectricity that may constitute the overhead costs of a manufacturing plant, for eample.

*escribing Profit from Sales

%uppose you are an entrepreneur who has 5ust purchased &, apples for I:. $f you ma"e &cents profit on each apple, you may want to find your total profit if you sell all the apples. Msethe equation # 0 p O s - c, where # 0 total profit, p 0 profit on each sale, s 0 number of sales andc 0 your costs. =lugging the values into the equation, you have #0 .&(&,) - : 0 I:. $f, onthe other hand, you only sell G apples, the equation becomes # 0 .&(G) - : 0 - I&. >ouwould be I& in the hole.

anipulating Linear Equations 'rea0even alculation

8emember that an equation is an epression of equality, so you can perform any type ofarithmetic operation on both sides of the equation. %uppose in the previous eample you want to"now how many apples you need to sell to brea" even. %o now your un"nown value is not #.>ou are setting that equal to ero to try to find !s,! the total number of sales. %tart with the sameequation, # 0 p O s - c. =lug in the values that you "now+ 0 .&(s) - :. Dow isolate theun"nown variable. %tart by adding : to both sides of the equation+ : 0 .&(s). Dow divide bothsides by .&. >ou?ll get : 0 s. >ou need to sell : apples to brea" even.


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alculating osts

When calculating anything of which you will buy more than one and which has some etra fiedcost attached, you will use a linear equation. $f you are renting a car for a day your total cost willbe the day?s rental plus each gallon of fuel used multiplied by the cost of the fuel. $f you have a

phone contract you may pay a fied line rental plus the number of minutes you called multipliedby the cost of each minute. Msing linear equations can also help you to ma"e the mosteconomical choice of tariff. A tariff with a high fied cost but a low unit cost will prove cheaperthan a tariff which offers the opposite only after a certain number of units have been used. Alinear equation will allow you to determine this number.

onverting 'etween 1nits

;onverting between unit types -- whether a conversion from one currency to another, from theimperial to the metric system of measurement, or from Hahrenheit to ;entigrade -- requires alinear equation. $f one dollar equaled two 7ritish pounds, then two dollars would be four pounds

and so on. A steady change in one variable leads to a steady change in the other. Nne mile isequal to &.9B "ilometers, so "0 m &.9B, again creating a steady relationship between the twounits which can be illustrated by a linear equation.

2eights and easures

%ay in a bread production factory, a recipe calls for & grams of flour, but you can only weighin ounces. >ou use a mathematical formula to convert grams to ounces. Nr you measure thedriveway to figure out how much concrete you will need to pave it. 7udgeting, investing,sewing, coo"ing--math is everywhere.

Simple Sample

A cleaning contractor has two employees, A and 7, who are available to clean a particular officebuilding. Hrom prior eperience, their manager "nows that A can clean this comple in : hours.Also, A and 7 wor"ing simultaneously--A from the bottom floors up, 7 from the top floorsdown--can get it done in .: hours. Kow long would it ta"e 7 to do the 5ob alone@#he linear equation that would come in handy here is &:(.:) 1 &t(.:) 0 &.4ultiplying both sides by :t yields+ .:t 1 (.:)(:) 0 :t, wor"ing that through yields a t of &&.9Chours. #he contractor should probably fire 7 and hire more As.

Standard *efinition

#he eample &:(.:) 1 &t(.:) 0 & is a linear equation by the standard definition, which meansit is an algebraic equation in which there is no variable of higher than the first degree.$t isn?t an especially interesting linear equation, though, because it only has one variable. We"now everything about employee A going in, so the only variable t was that representing ourdesideratum, 7?s time.


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7oth mathematical interest and business applications increase when we add another variable.Kowever, we will stic" to the rule that only first-power variables, which graph as straight lines,are allowed.

%llocating osts 'etween *epartments

%uppose a particular business has both an engineering department (6) and a generalmanufacturing plant (

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