CH 20 SOLVING FORMULAS

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CH 20 SOLVING FORMULAS 179 Itrodutio S olvig equtios suh s 2 + 7 20 is oviousl the orerstoe of lger.

For emple, if ou wted to fid the temperture i the gs lw PV RT, ou might hve to solve this equtio for T.

1 CH 20 SOLVING FORMULAS 179 Itrodutio S olvig equtios suh s is oviousl the orerstoe of lger. But i siee, usiess, d omputers it is lso eessr to solve equtios tht might hve vriet of letters i them. For emple, if ou wted to fid the temperture i the gs lw PV RT, ou might hve to solve this equtio for T. A formul (lled literl equtio m) is just equtio with two or more vriles i it, d is desiged to omplish somethig speifi. The equtio for the perimeter of retgle, P 2l + 2w, is emple of formul. If we eeded to solve for the width i this retgle formul, we would hve to isolte the vrile w usig the sme equtio-solvig tehiques we ve just studied. Review of Solvig Equtios Some studets fid the solvig of formuls to e ver strt, while others osider it muh esier th solvig regulr equtios. Let s egi this hpter with quik review of solvig oe-step equtios. To solve the equtio , we reogize tht 3 hs ee dded to the ukow. To remove the 3 -- tht is, to udo the dditio -- we sutrt 3 from eh side of the equtio. This isoltes, or solves for, the, d we get 7. Similrl, to solve the equtio 10 27, we dd 10 to eh side of the equtio, resultig i solutio of 37. To isolte the vrile i the equtio 9w 45, we otie tht multiplitio ids the 9 with the w. To remove the 9, therefore, we divide eh side of the equtio 9, ieldig vlue of 5 for w. Ch 20 Solvig Formuls

2 180 For our lst review emple i this setio, to solve the equtio we re oliged to multipl eh side of the equtio 12, sie multiplig udoes dividig. Ad so 288. I the followig setios we lso isolte vriles, ut ow we hve to remove other vriles (ot just umers) i order to isolte the oe we wt. Sie there s o rithmeti ivolved -- just lger oepts -- ou see wh some studets thik this is ot so d, fter ll. Perhps with little prtie ou ll feel the sme w. Oe-Step Emples EXAMPLE 1: Solve for : + Solutio: To isolte the we eed to remove the. Sie the is eig dded to the, we remove it sutrtig it from eh side of the equtio, thus isoltig the : + (the origil formul) + (sutrt from eh side) (simplif) EXAMPLE 2: Solve for u: u w A Solutio: To isolte the u, we eed to remove the w. Sie the w is eig sutrted from the u, we remove it ddig w to eh side of the equtio: u w A (the origil formul) u w + w A + w (dd w to eh side) u A + w (simplif) Ch 20 Solvig Formuls

3 181 EXAMPLE 3: Solve for : R Solutio: To isolte the we eed to remove the. Sie the opertio etwee the d the is multiplitio, we remove the dividig eh side of the equtio : R (the origil formul) R (divide eh side ) R (simplif) EXAMPLE 4: Solve for W: W Solutio: To isolte the W we eed to remove the. Sie W is eig divided, we remove the multiplig eh side of the equtio : W (the origil formul) W (multipl eh side ) W (ross-el the s) W (simplif) EXAMPLE 5: Solve for : ( z) Q Solutio: To isolte the we eed to remove the qutit z. We sk, Wht is the opertio oetig the with the z? It s multiplitio, so we use divisio to reverse the opertio d thus isolte the : ( z) Q (the origil formul) ( z) Q z z (divide eh side z) Q z (simplif) Ch 20 Solvig Formuls

4 182 EXAMPLE 6: Solve for the temperture T i the gs lw PV RT metioed i the Itrodutio. Solutio: PV RT (the origil gs lw) PV R PV R RT R T (divide eh side R) (simplif the right side) T PV R (reverse the equtio) EXAMPLE 7: Solve eh formul for : A. Now omes ew issue -- we kow tht we eed to multipl eh side of the equtio (i order to isolte the ). Multiplig the left side is es; it ross-els with the, levig just, the ukow. But how do we idite tht the right-hd side of the equtio, the qutit +, ll of it, must e multiplied? We put pretheses roud the qutit +, tht s how: ote the pretheses ( ) Simplifig the left side of the equtio gives. Simplifig the right side gives simpl ( + ). Thus, ( + ), or, the ommuttive propert for multiplitio, ( + ), d we re doe. Note: Aother w to write the fil solutio is to distriute d get +. But this is ot eessr, sie the gol of this hpter is to ler how to isolte vriles, ot simplif swers. Ch 20 Solvig Formuls

5 183 B. z w Agi, we re trig to isolte the, d gi we will omplish this multiplig eh side of the equtio the deomitor, i this se z. Rememerig the use of pretheses (or rkets), we get C. u e w z z z, d our fil swer is w[ z], or w( z) Multiplig eh side of the equtio e produes ( u) e e e ( + u)( e) Homework 1. Solve eh formul for :. +. d. z d. T e. + m f. z L g. R + w h. i. m j. + k. + W l. z m. + w. o. + d 2. Solve eh formul for :. ( ) d. (Q + R) S. d. d e. mg P f. ( ) A 1 2 ( r r ) T 1 2 g. M h. ( ) z i. (w u) j. z k. W l. Q (g h) Ch 20 Solvig Formuls

6 Solve eh formul for :.. d e. d. t d e. w z f. z A g. w 7 h. Q i. w j. g h k k. L M N P l. u w z m. R Q T. m g h o. w u p. w q. u w deh Two-Step Emples Let s review stdrd two-step equtio. If we solve the equtio for, we rememer the dilemm of deidig whih to rid ourselves of first, the 3 or the 17. We greed tht reversig the Order of Opertios ws the seret. So, from the s poit of view, it ws multiplied 3 first, d the 17 ws sutrted. Reversig this sequee mes we first dd 17 to eh side, givig 3 99; the we divide eh side 3 to get the vlue 33. Well, it s eve esier with letters [I hope!]. EXAMPLE 8: Solve for : Solutio: The Order of Opertios tells us tht first the d the were multiplied, d the ws sutrted. To isolte the we eed to reverse the Order of Opertios: dd to eh side, d the divide eh side : (the origil formul) Ch 20 Solvig Formuls

7 (dd to eh side) + (simplif) (divide eh side ) (simplif) Note: It s ommo to use lphetil order whe writig vriles, so the swer lso e writte. EXAMPLE 9: Solve for : e d Solutio: The hs ee divided, d the e hs ee dded. Reversig these opertios produes the followig steps: e d (the origil formul) e d e e (sutrt e from eh side) d e (simplif) Now rell from the previous setio how we isolte the. We eed to multipl eh side of the equtio, d we eed to put pretheses roud the qutit d e whe we do. ( d e) (multipl eh side -- otie the pretheses) ( d e) (simplif) Homework 4. Solve eh formul for :. +. d R. + d. m Q e. e h f. Ch 20 Solvig Formuls

8 186 g. + d h. p i. k L d j. d k. + d e l. m. t + R. h h o. p. 2 + d q. 7 r. j N e 5. Solve eh formul for : Emple: Solve for : Solutio: (sutrt 3 from eh side) (sutrt 1 from eh side) (divide eh side 5) d e f g h i j k l Multi-Step Emples EXAMPLE 10: Solve for : T Solutio: To solve this formul for, we must remove the T, the, d the. At eh step determie the fil opertio, d perform the reverse opertio. Strt with the origil prolem: T The fil opertio is divisio. Remove the multiplig eh T ( ) side of the equtio : Ch 20 Solvig Formuls

9 187 Simplif eh side: T Now the fil opertio is the dditio of the. Remove the T sutrtig it from oth sides: Now divide oth sides T: T T T Simplif eh side, d we re doe: T EXAMPLE 11: Solve for : w + z e f Solutio: This formul hs the sme struture s the oe i the previous emple, eept tht the deomitor d the qutit o the right osist of two terms isted of oe. The ol thig to rememer is to use pretheses wherever pproprite. w e f z w z e f z z (origil formul) (multipl eh side + z) w (e f)( + z) (simplif) ( e f )( z) w (dd w to eh side) ( e f )( z) w (divide eh side ) EXAMPLE 12: Solve for R: R d Q Solutio: This might e the pproprite ple to show ou other pproh to solvig omplited formuls. The evetul steps we will tke will e etl s we ve doe i the previous emples. But some studets like to see the Order of Opertios epliitl umered; this helps them see wht opertio to udo t eh stge of the prolem. Ch 20 Solvig Formuls

10 188 Preted ou were R, the ukow. Let s list etl wht s ee doe to ou, usig the Order of Opertios: 1. ou were multiplied 2. ws the sutrted 3. the whole thig ws the divided 4. lst, d ws dded o To utgle this mess, we eed to reverse the Order of Opertios, d reverse the opertio t eh stge of the Order of Opertios: 1. sutrt d 2. multipl 3. dd 4. divide The origil formul: R d Q 1. Sutrt d from eh side: R d d Q d Simplif: R Q d 2. Multipl eh side : R ( Q d ) Simplif: R ( Q d) 3. Add to eh side: R ( Q d) Simplif: R ( Q d) 4. Divide eh side : R ( Q d) We mde it! ( Q d) R Ch 20 Solvig Formuls

11 189 Homework 6. Solve eh formul for :. d. g. j. d. R u w t h. 3 d w. T e. L M f. m w T k. z m. 7 u w. z p. s. v.. d d q. R u w w t. d d Z i. g h d L A T l. d Q o. w r. h p p q T e e L M u. d w. z A. S z. z g h L Q w T Solvig Rel Formuls Homework 7. Solve for m i the kilometers/miles formul k 1.61m. 8. Solve for k i the kilometers/miles formul k m Ch 20 Solvig Formuls

12 Solve for C i the temperture formul F 1.8C Solve for F i the temperture formul C 32 F Solve for L i the ssets/liilities/pitl formul A L + C. 12. Solve for R i the profit/reveue/epese formul P R E. 13. Solve for t i the diste/rte/time formul d rt. 14. Solve for m i the desit/mss/volume formul d m. V 15. Solve for S i the verge formul A S. 16. Solve for i i Ohm s Lw V ir. 17. Solve for R i the gs lw PV RT. 18. Solve for m i the potetil eerg formul P mgh. 19. Solve for m i Eistei s equtio E m Solve for l i the perimeter formul P 2l + 2w. 21. Solve for i the irle re formul A r 2. Fil Note Here s prtig ommet regrdig the solvig of formul: The fil swer ever oti the vrile ou re solvig for. Ch 20 Solvig Formuls

13 191 For emple, fil solutio suh s + is outright isit. Here s wh: This solutio ss tht to determie the vlue of, we eed to kow the vlues of,,, d! A ftl se of irulr resoig, ideed. Ad ow for somethig iroi: I omputer lguges the sttemet + 3 is perfetl vlid d tremedousl useful (it merel ireses the vlue of 3.) But it is ot equtio we re trig to solve, so it does t violte the rule ove. Review Prolems 22. Solve for w i the perimeter formul P 2l + 2w. 23. Solve for w i the re formul A lw. 24. Solve for r i the irumferee formul C 2r. 25. Solve for d i the rdius formul 26. Solve for A i the formul 27. Solve eh formul for : A r 2 r d. 2.. d. 2 + R. d. N e. ( + z) f. d u g. m h. Q i. Q R j. ( ) A k. e m 1 2 l. m Your eighor solves the formul + for, d omes up with solutio of. Your ommets, plese. Ch 20 Solvig Formuls

14 192 Solutios 1... d. z +, or + z d. TL, or LT e. m f. z g. W + R h. i. + m j. k. W + l. z, or z m. w. + o. d 2.. d d. d g. M j. z. S Q R e. P mg h. z k. W. f. i. l. A T r r 1 2 w u Q g h ( + ). (d e). ( + ) d. t( + d) e. ( w + z) f. A( z) g. (w + 7) h. Q( ) i. w( + ) j. k( + g h) k. P(L + M N) l. ( + z)(u + w) m. (Q + T)( R). (g h)(m ) o. (w + u)( + ) p. ( + w)( + + ) q. (d e + h)(u + w ) 4.. d. Q m. R d. e. (h + e), or (h + e) Ch 20 Solvig Formuls

15 193 f. + g. d h. pd i. (L k), or (L k) j. d k. e d l. m. R t. 0 o. p. d 2 q. e 7 r. (N + j), or (N + j) d. 7 3 e f g. 2 1 h i j l k dw. A 3 T d. R( + d) e. ( M L) f. ( w)( ) m g. w u h. ZT i. LT d j. t k. ( g h)( z) ( l. ) ( ) or m. z w u d( ). 7 o. ( )( p q) p p. d q. wd h r. e e T s. R( d) t. ( M L) ( w)( ) u. Ch 20 Solvig Formuls

16 194 v.. d w u w. w z. AS z. ( g h)( z) TQ L 7. m k 8. k 1.61m C F F 1.8C L A C 12. R P + E 13. t d r 16. i V R m E w P 2l m dv 15. S A 17. R PV T l 23. A w l 25. d 2r 26. A r 2 P 2w m P gh 21. A 2 r 24. r C d +. R. 2 d. Nu e. f. ( d) z g. (m )( + ) h. ( + Q)( Q) i. R( + ) j. m. A k. (m + e) l. 28. Whe solvig formul for, the ol letter tht t possil e i the fil swer is itself. For oe thig, solvig formul for mes to isolte it. How isolted is it whe there s o eh side of the solutio? Seod, the ogus solutio eds up i irulr resoig. Aordig to our eighor s solutio, to fid the vlue of we would eed the vlues of,, d, d! Tht is, to fid the vlue of, we would eed the vlue of. Tht s rz. Ch 20 Solvig Formuls

17 195 To d Beod de gh k f g Solve for : w z Ch 20 Solvig Formuls

18 196 Tht is wht lerig is. You suddel uderstd somethig ou ve uderstood ll our life, ut i ew w. Doris Lessig Ch 20 Solvig Formuls

CH 19 SOLVING FORMULAS

1 CH 19 SOLVING FORMULAS INTRODUCTION S olvig equtios suh s 2 + 7 20 is oviousl the orerstoe of lger. But i siee, usiess, d omputers it is lso eessr to solve equtios tht might hve vriet of letters i them.