MAT 5 Projet 3: Usig Idetities to Rewrite Expressios Wldis I lger, equtios tht desrie properties or ptters re ofte lled idetities. Idetities desrie expressio e repled with equl or equivlet expressio tht hs differet form. So, for exmple, osider this idetity: + = +. This mes tht for y two thigs tht we re ddig together (umers, simple expressios, more omplex expressios ythig tht the order of opertios llows us to group together), we lwys swith the order d the result will still e equl to the origil expressio. The tul letters d re t importt. We ould lso write this idetity s x + y = y + x or s p + q = q + p. Exmple: A) Let s rewrite + = + usig the letter p for d the letter q for : = p, = q + = + + = ( ) + ( ) (p) + (q) = (q) + (p) So we hve p + q = q + p. Notie tht this hs the sme struture s the origil idetity + = +. Usig idetities to rewrite expressios Idetities llow us to rewrite expressios y replig expressio with differet equivlet expressio (tht my e simpler or more useful for grphig or solvig somethig lter). Tehilly we ever simply hge or move somethig i expressio. Tehilly, we oly rewrite expressios y replig some or ll of the expressio with equivlet oe. Defiitio: Two expressios re equivlet if d oly if they will lwys e equl, for ALL possile vlues of the vrile(s). (If expressios oti multiple vriles, this mes for y possile omitio of vrile vlues.) Exmples: For eh of these expressios, we eed to idetify wht is tkig the ple of the vrious vriles i the idetity. Just like i previous projets, wht tkes the ple of the vriles might ot just e umers, ut ould e more omplex expressios tht themselves oti vriles or multiple terms. We lso hve to e reful tht whe hoosig groupigs withi expressio, we lwys follow the order of opertios. So, s we dd pretheses to group thigs together, we hve to e sure tht we oly dd pretheses whe it does t hge the struture imposed y the order of opertios. B) Use the idetity + = + to reple the origil expressio with equivlet expressio tht hs differet form: x + 3y = x, = 3y + = + (x) + + (3y) = = (3y) + + (x) + 33 = 33 +
C) Use the idetity + = + (wheever 0) to reple the origil expressio with equivlet expressio tht hs differet form: x + 4y = x, = 4y, = (Sie 0, we kow tht 0) + = + + = ( ) + ( ) (x ) +(4y) = (x ) + (4y) () () () x +44 = x + 44 D) Use the idetity x = x (whe is positive whole umer) to reple the origil expressio with mmmm ttttt equivlet expressio tht hs differet form: (3xx 4) 3 Importt poit! You my otie i this prolem tht we hve x i the idetity d tht we lso hve x i the expressio. This e ofusig. These two x s re NOT the sme, euse they re omig from two totlly differet otexts. I the idetity, the x is just stdig i for ANY expressio; wheres i the expressio, the x my hve prtiulr vlue or my represet prtiulr thig tht is vryig (for exmple: time, or diste). I this se, it might e esier to first rewrite the idetity with differet letter isted of usig x. For exmple, we ould rewrite the idetity usig the letter i ple of x to void the ofusio: x = x x mmmm ttttt = mmmm ttttt = 3xx 4, = 3 (Sie 3 is positive whole umer, we kow tht is positive whole umer) = ( ) ( ) mmmm ttttt (3) (3xx 4) = (3xx 4) (3xx 4) (3) mmmm ttttt (3) (3xx 4) = (3xx 4) (3xx 4) (3xx 4) (3) mmmm ttttt (333 4) 3 = (333 4) (333 4) (333 4) Use the idetity x 0 = (wheever x 0) to reple the origil expressio with equivlet expressio tht hs differet form: (xy 3y) 0 (wheee xy 3y 0) We ould rewrite the idetity x 0 = (wheever x 0) with isted of x to void the ofusio etwee the two x s (oe i the idetity d other i the expressio): x 0 = (wheever x 0) 0 = (wheever 0) = xy 3y 0 = 0 = ( llll 0) 0 (xy 3y) = ( ww kkkk th xy 3y 0) (y 33) 0 =
Now you try! For eh of the followig prolems, use the give idetity to rewrite the give expressio i differet equivlet form. Use the exmples ove s model, writig out ll of the sme steps.. Use the idetity = + to reple the origil expressio with equivlet expressio tht hs differet form: x 3y. Use the idetity = (wheever 0) to reple the origil expressio with equivlet expressio tht hs differet form: (x )(4y) 3. Use the idetity = x + + x (whe is positive whole umer) to reple the origil expressio with mmmm ttttt equivlet expressio tht hs differet form: 3(xx 4) 4. Use the idetity ( + ) = + to reple the origil expressio with equivlet expressio tht hs differet form: (x )(3x + 7)
Does it mtter whether expressio hs the form of the right or the left side of idetity? You my hve otied tht i ll the exmples so fr, we hve tke somethig tht hs the struture of the left side of the idetity d repled it with whtever is o the right side of the idetity. But we lso do the reverse, euse: If two thigs re equl, we lwys reple oe with the other! It does t mtter whih side of equtio is writte o the left or the right the equls sig i the middle just tells us tht oth sides re equl. Exmples: F) Use the idetity x = x (wheever x 0) to reple the origil expressio with equivlet expressio tht hs differet form: 8 (3p r 0) 3p r This expressio hs the struture of the right side of the idetity x = x rther th the left side. If it is helpful, we ould rewrite the idetity to look like this: = x x (wheever x 0), euse this is just other wy of syig extly the sme thig tht the two sides re equl. We ow use this idetity to rewrite our expressio x = 3p r, = 8 x = x x = x x (8) (3p r) So x = (3p r) (8) = 33 r 8 33 r 8 3p r 8 : (We kow tht x 0 euse 3p r 0.) G) Use the idetity ( + ) = + to reple the origil expressio with equivlet expressio tht hs differet form: x(3x 3 y 5 ) + x(3y 4x) This expressio hs the struture of the right side of the idetity ( + ) = + rther th the left side. If it is helpful, we ould rewrite the idetity to look like this: + = ( + ), euse this is just other wy of syig extly the sme thig tht the two sides re equl. We ow use this idetity to rewrite our expressio x(3x 3 y 5 ) + x(3y 4x): = x, = 3x 3 y 5, = 3y 4x + = ( + ) (x) + (3x 3 y 5 ) + (x) = (3y 4x) + ( ) = (x) (3x 3 y 5 ) + (3y 4x) So ()3x 3 y 5 + ()(33 44) = () 3x 3 y 5 + (33 44)
Now you try! For eh of the followig prolems, use the give idetity to rewrite the give expressio i differet equivlet form. Use the exmples ove s model, writig out ll of the sme steps. 5. Use the idetity ( + ) = + to reple the origil expressio with equivlet expressio tht hs differet form: 3xx( 3 ) + 3xx( ) 6. Use the idetity + = + (wheever 0) to reple the origil expressio with equivlet expressio tht hs differet form: 9x + 3 x 3 3 7. Use the idetity = tht hs differet form: (wheever 0) to reple the origil expressio with equivlet expressio ppp pq r 8. Use the idetity + = ( + ) to reple the origil expressio with equivlet expressio tht hs differet form: (x + 7)(xx )