Mth 95 Notes Module 1 Spring Section 4.1- Solving Systems of Linear Equations in Two Variables by Graphing, Substitution, and Elimination

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1 Mth 9 Notes Module Sprig 4 Sectio 4.- Solvig Sstems of Liear Equatios i Two Variales Graphig, Sustitutio, ad Elimiatio A Solutio to a Sstem of Two (or more) Liear Equatios is the commo poit(s) of itersectio ad produces true math seteces whe sustituted ito oth equatios Determie sustitutio whether (-, ) ad (-6, -6) are solutios to the sstem. Numer of Possile Solutios to a sstem of Liear Equatios A B C solutio(s) solutio(s) of solutio(s) Slopes are Slopes are the Slopes are the Y-itercepts are Y-itercepts are Y-itercepts are the A cosistet sstem has at least oe solutio. A icosistet sstem has o solutio. Which of the sstems aove are cosistet? Icosistet? Idepedet equatios are differet. Depedet equatios are the same (slope ad - itercept). Which equatios aove are idepedet? Depedet? To tell how ma solutios a sstem of equatios has, trasform oth equatios ito slope-itercept form ad eamie their m-values ad -values. Tell how ma solutios each sstem has Chapters 4 ad

2 Mth 9 Notes Module Sprig 4 Methods of Solvig a Sstem of Equatios Method : Solvig Graphig: good for approimatio, a good visual of the situatio, also a good check for our smolic solutios. Steps: ) Graph oth equatios o the same coordiate plae. ) Name the poit(s) of itersectio ) Check our solutio algeraicall sustitutig the poit of itersectio ito oth equatios or trasformig oth equatios ito slope itercept form ad eamiig m ad for ifiitel ma solutios ad o solutio. Solve each sstem graphig. 6 6 Solutio Solutio Solutio Show a algeraic check for our solutio to the first sstem aove. Check the other sstems graphig them o our calculator. Method : Solvig Sustitutio (works well whe the coefficiet of or is oe.) Steps: ) Choose oe of the equatios ad solve it for oe of the variales. ) Sustitute what ou get ito the other equatio ad solve. ) Check our solutio. Use sustitutio to solve the followig sstems of liear equatios 6 6 Solutio Solutio Solutio Chapters 4 ad

3 Mth 9 Notes Module Sprig 4 Applicatios: Use sustitutio to solve the followig: (Break Eve prolem) Fid the umer of uits,, that must e sold to reak eve give the Cost (C) ad Reveue(R) fuctios for a usiess to e: C( ), R( ) Method : Elimiatio Steps: ) Get oth equatios i stadard form. Be sure the same variales lie up uder each other ad the equal sigs are uder each other. ) Multipl or divide oe or oth the equatios to get the same coefficiet ut opposite sigs o oe of the variales. ) Comie the like variales ad costats i a vertical maer. Oe of the variales should e elimiated. 4) Solve for the remaiig variale. ) Check our solutio. Use the elimiatio method to solve the followig: 7 Solutio 47 Solutio 4 Solutio 6 6 Solutio Chapters 4 ad

4 Mth 9 Notes Module Sprig 4 4 Solutio What does it mea whe oth m variales disappear i the process? Sectio 4. Sstems of Liear Equatios ad Prolem Solvig Applicatios: Begi defiig our variales. The write a sstem of equatios ad use the elimiatio or sustitutio method to solve each prolem. (Miture Prolem): Oe umer is two less tha a secod umer. Twice the first is 4 more tha times the secod. Fid the umers. (Miture Prolem): A grocer mies two grades of coffee that sell for $6 ad $8 a poud. He has a i that holds pouds of coffee. How much of each must he u to make a miture that sells for $7. a poud? Chapters 4 ad 4

5 Mth 9 Notes Module Sprig 4 (Miture prolem with ivestmets): A sum of $ is ivested for oe ear. Some of the moe is ivested at 6% ad the remaider at %. How much was ivested at each percet if the total iterest icome from the two ivestmets is $8? (Aother miture prolem with percets): A pharmacist eed milliliters of a % Pheoarital solutio ut has ol % ad % Pheoarital solutios availale. Fid how ma milliliters of each he should mi to get the desired solutio. (A motio prolem where curret affects the rate) A oat travels 8 miles dow a river i hour. A retur trip agaist the curret takes. hours. Fid the average speed of the oat i still water ad the rate of the curret. (distace = rate time) Chapters 4 ad

6 Mth 9 Notes Module Sprig 4 Sectio. Iteger Epoets I the epressio 4, 4 is the ad is the. is repeated as a times, so Properties of Epoets: ) Product Rule 4 meas 4 4 = m m Whe ou multipl two umers with the same keep the ad add the 4 ( )( ) z z (4 )( )( ) ) Zero Epoets For a ozero umer a, ut a =, so is = ad =, ( 8) ( ) ( ) ) Quotiet Rule for Epoets m m Whe ou divide two umers with the same, sutract the epoets ( top epoet ottom epoet). 4cd 4cd ) Negative Epoets (does t chage the sig of the coefficiet) 7 Chapters 4 ad 6

7 Mth 9 Notes Module Sprig 4 Use properties of epoets to evaluate each epressio. Give all aswers with positive epoets. 7 ( ) 4 r t rt r w t ( z) Review Sectio. Product Rule for Epoets 4 ( )( ) Quotiet Rule for Epoets c c Zero Epoets ( 4 ) Negative Epoets m Sectio. - More Work with Properties of Epoets ) Raisig Powers to Powers - Multipl the epoets. m m ( ) ( ) ( ) Chapters 4 ad 7

8 Mth 9 Notes Module Sprig 4 6) Raisig Products to Powers - Take each factor to that power. ( a) a ( ) 4 m a 4 7 7) Raisig Quotiets to Powers - Take the umerator ad the deomiator to that power. a a m 4 7 Simplif the followig epressios completel. Write aswers usig positive epoets ol cd 9cd c d 4 c r t r t 6 Chapters 4 ad 8