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pictured below. Simplify your answer as much as possible.WLWLssssSolution.The perimeter of any polygonal figure is the sum of the lengths ofits sides.a) To find the perimeterPof the rectangle, sum its four sides.P=L+W+L+W.Combine like terms.P= 2L+ 2W.b) To find the perimeterPof the square, sum its four sides.P=s+s+s+s.Combine like terms.P= 4s.Answer:P= 6xSometimes it is useful to replace a variable with an expression containinganother variable.You Try It!EXAMPLE 11.The length of a rectangle is three feet longer than twice itsThe lengthLof a rectangleis 5 meters longer than twiceits widthW. Find theperimeterPof the rectanglein terms of its widthW.width. Find the perimeterPof the rectangle in terms of its width alone.Solution.From the previous problem, the perimeter of the rectangle is givenbyP= 2L+ 2W,(3.1)whereLandWare the length and width of the rectangle, respectively. Thisequation gives the perimeter in terms of its length and width, but we’re askedto get the perimeter in terms of the width alone.However, we’re also given the fact that the length is three feet longer thantwice the width.
204CHAPTER 3.THE FUNDAMENTALS OF ALGEBRALengthisThreeFeetlonger thanTwice theWidthL=3+2WBecauseL= 3+2W, we can replaceLwith 3+2Win the perimeterequation 3.1.P= 2L+ 2WP= 2(3 + 2W) + 2WUse the distributive property, then combine like terms.P= 6 + 4W+ 2WP= 6 + 6W.This last equation gives the perimeterPin terms of the widthWalone.Answer:P= 6W+ 10You Try It!EXAMPLE 12.The width of a rectangle is two feet less than its length.The widthWof a rectangleis 5 feet less than twice itswidthL. Find the perimeterPof the rectangle in termsof its lengthL.Find the perimeterPof the rectangle in terms of its length alone.Solution.Again, the perimeter of a rectangle is given by the equationP= 2L+ 2W,(3.2)whereLandWare the length and width of the rectangle, respectively. Thisequation gives the perimeter in terms of its length and width, but we’re askedto get the perimeter in terms of the length alone.However, we’re also given the fact that the width is two feet less than thelength.WidthisLengthminusTwo feetW=L-2BecauseW=L-2, we can replaceWwithL-2 in the perimeterequation 3.2.P= 2L+ 2WP= 2L+ 2(L-2)Use the distributive property, then combine like terms.P= 2L+ 2L-4P= 4L-4.This last equation gives the perimeterPin terms of the lengthLalone.Answer:P= 6L-10
3.4.COMBINING LIKE TERMS205❧❧❧Exercises❧❧❧In Exercises1-16, combine like terms by first using the distributive property to factor out the commonvariable part, and then simplifying.1. 17xy2+ 18xy2+ 20xy22. 13xy-3xy+xy3.-8xy2-3xy2-10xy24.-12xy-2xy+ 10xy5. 4xy-20xy6.-7y3+ 15y37. 12r-12r8. 16s-5s9.-11x-13x+ 8x10.-9r-10r+ 3r11.-5q+ 7q12. 17n+ 15n13.r-13r-7r14. 19m+m+ 15m15. 3x3-18x316. 13x2y+ 2x2yIn Exercises17-32, combine like terms by first rearranging the terms, then using the distributiveproperty to factor out the common variable part, and then simplifying.