Review for College Algebra Final Exam


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1 Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i two variables Vocabulary: system of (liear) equatios, solutio of a system of equatios, cosistet system, icosistet system, graphical method, substitutio method, elimiatio method, equivalet equatios A solutio (x, y) of a system of two equatios is a ordered pair which makes both equatios true. Graphical method: Each liear equatio defies a lie i the plae. Graph the lie for each equatio i the system ad fid the itersectio (overlap) of the lies. The itersectio could be lie (if the two lies are the same), a poit (most commo), or othig (if the lies are parallel but ot the same). If a system of equatios has at least oe solutio, it is cosistet. If ot, it is icosistet. Substitutio method: I oe of the equatios, solve for oe variable (say x) i terms of the other (say y) ad use the result to substitute for x i the other equatio. This other equatio will ow ivolve oly y s, ad you ca solve for y. Plug this yvalue back ito the equatio you solved for x (this is called backsubstitutio) i order to fid x. Of course, x ad y ca be switched i this whole process. Elimiatio method: Multiply oe or both equatios by a costat to get equivalet equatios for which either the x or the ycoefficiets are egatives of each other. Add the equatios to elimiate oe variable. Solve for the other variable, ad the backsubstitute ito oe of the origial equatios to fid the variable which was elimiated. 5.2 Systems of equatios i three variables To solve a system of three liear equatios i three variables, choose two pairs of equatios from the three equatios, ad elimiate the same variable from each pair usig the elimiatio method from 5.1. Now you have a system of two liear equatios i two variables. Use the elimiate method agai to elimiate aother variable ad solve for the value of the oe remaiig variable. Fially, backsubstitute ito oe of the earlier twovariable equatios to fid the value of a secod variable, ad the backsubstitute both values ito oe of the origial equatios to fid the value of the third variable. 5.7 Systems of iequalities ad liear programmig Vocabulary: liear iequality, halfplae, system of liear iequalities, objective fuctio, costraits, regio of feasible solutios
2 Graphig a liear iequality A liear iequality is a iequality that ca be writte i the form Ax + By < C, with A ad B ot both zero. The < sig may be replaced by >,, or. The iequality may also be give i slopeitercept form, such as y < mx + b. To graph the solutio set of a liear iequality: 1. Replace the iequality symbol with a equals sig ad draw the graph of the resultig liear equatio. Use a dotted lie if the symbol is < or > ad a solid lie if the symbol is or. The dotted lie meas that the poits o the lie are ot solutios, while the solid lie meas that the poits o the lie are solutios. 2. To determie which side (or halfplae) of the lie cotais solutios to the iequality, test whether a poit o oe side of the lie is a solutio. If it is, the shade that side of the lie. If it is ot, shade the other side of the lie. Graphig a system of liear iequalities A system of iequalities cosists of two or more iequalities cosidered simultaeously. For example: x y 5 2x + 3y > 0 The solutio set of a system is the set of poits which are solutios to every iequality i the system. To graph the solutio set of a system of liear iequalities: Graph the solutio set of each iequality o the same set of axes. The overlap of these solutios sets is the solutio set for the whole system. Istead of shadig the halfplae for each iequality, it is cleaer to draw small arrows idicatig which side would be shaded. The, usig this iformatio, you ca shade oly the solutio set for the etire system of equatios. A poit o more tha oe of the boudary lies is icluded i the fial solutio set if ad oly if all of the boudary lies cotaiig it are solid lies. Liear programmig To fid the maximum or miimum value of a liear objective fuctio subject to a set of liear costraits: 1. Graph the regio of feasible solutios (the solutio set for the system of costraits). 2. Fid the coordiates of the vertices of this regio of feasible solutios. 3. Evaluate the objective fuctio at each vertex ad fid the largest or smallest value.
3 7.1 Sequeces ad series Vocabulary: sequece, ifiite/fiite sequece, terms of a sequece, geeral term, alteratig sequece, series, ifiite/fiite series, partial sum, sigma/summatio otatio, idex of summatio, explicit defiitio, recursive defiitio Give a explicit or a recursive defiitio of a sequece, fid particular terms of the sequece directly from the defiitio. Give particular terms of a sequece, fid a explicit or a recursive defiitio of the sequece. Calculate partial sums by addig up the first few terms of a sequece. Evaluate expressios writte usig summatio otatio. Write expressios usig summatio otatio. 7.2 Arithmetic sequeces ad series Vocabulary: arithmetic sequece, commo differece, th term, arithmetic series Fid a recursive formula for a arithmetic sequece usig the first term a 1 ad the commo differece d: { a 1 = value of a 1 = a + d a +1 Fid a explicit formula for a arithmetic sequece usig the first term a 1 ad the commo differece d: a = a 1 + ( 1)d. Give ay two terms of a arithmetic sequece, fid the explicit formula for the sequece ad use it to determie ay other term of the sequece. Give the value of a term i a arithmetic sequece, determie which term it is. Fid the sum of a fiite arithmetic series usig the formula: ( ) a1 + a S = a k =. 2 Use arithmetic sequeces ad series to solve applied problems. 7.3 Geometric sequeces ad series Vocabulary: geometric sequece, commo ratio, geometric series, ifiite geometric series, limit Fid a recursive formula for a geometric sequece usig the first term a 1 ad the commo ratio r: { a 1 = value of a 1 = a r a +1
4 Fid a explicit formula for a geometric sequece usig the first term a 1 ad the commo ratio r: a = a 1 r 1. Give ay two terms of a geometric sequece, fid the explicit formula for the sequece ad use it to determie ay other term of the sequece. Give the value of a term i a geometric sequece, determie which term it is. Fid the sum of a fiite geometric series usig the formula: S = ( ) 1 r a k = a 1. 1 r Fid the sum of a ifiite geometric series usig the formula: S = a k = a 1 1 r. Write a ifiitely repeatig decimal as a fractio by expressig it as a ifiite geometric series ad the usig the sum formula above. Use geometric sequeces ad series to solve applied problems. 7.5 Combiatorics: permutatios Vocabulary: permutatio, the fudametal coutig priciple, combied actio, with/without repetitio, idetical/odistiguishable objects A permutatio is a ordered arragemet. Fid the umber of permutatios of objects usig the formula: P =! = ( 1)( 2) (3)(2)(1). Fid the umber of permutatios of objects take k at a time usig the formula: P k =! = ( 1)( 2) ( (k 1)). ( k)! If a combied actio ca be performed as a sequece of k steps, ad the ith step ca be performed i i ways, the the combied actio ca be performed i ways k Give differet letters, the umber of kletter codes that ca be formed from the letters, with repetitio allowed, is k.
5 Suppose we have k differet types of objects, with i objects of type i, for each i. Suppose there are = k total objects. The the umber of arragemets of all objects is:! 1! 2! k!. Apply these priciples to solve various coutig problems. The formulas will ot be give to you. It will be easier to remember them if you uderstad why they are true. 7.6 Combiatorics: combiatios Vocabulary: combiatio A combiatio is a uordered selectio or choice. Fid the umber of combiatios of objects take k at a time usig the formula: ( ) C k = = P k! = k kp k k!( k)!. Apply this formula, alog with the fudametal coutig priciple, to solve various coutig problems. The formula will ot be give to you. It will be easier to remember it if you uderstad why it is true. Choosig k people out of people to be o your committee is equivalet to choosig ( k) people out of people ot to be o your committee, hece: ( ) ( ) =. k k We ca cout the umber of committees of ay size which ca formed out of people i two differet ways, yieldig two expressios which must be equal: ( ) ( ) ( ) ( ) =
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