4x 2-2x 3. Chapter 2: Variable Expressions

Transcription

1 Chpter : Vrible Expressions Expressions (contin no sign) : An expressionis one or numbers or vribles hving some mthemticl opertions done on them. Numericl Expressions: (4) / Expressions cn be evluted or simplified: cn be simplified to 8 This just mens, Whtever you see, do Algebric Expressions: x + 5 3x in Algebr, it is implied tht 3x mens 3 times x. The proper wy to write the product of number nd vrible is to lwys write the number to the left of the vrible. x times 5 5x When numbers re multiplied by vribles, they re given specil nme, coefficient. 5 is the coefficient of 5x. The quntities being dded in n lgebric expression re clled theterms. If term is vrible or combintion of vribles multiplied by numbers, it is clled vrible term. Numbers tht re just dded in the expression re clled constnt terms. Exmple: 3 terms 4x - x 3 vrible terms constnt term Hey! x nd 3 ren t being dded, they re being subtrcted! If we write this expression s 4x + -x + -3, then we hve ll term being dded, nd the constnt term is -3 nd the vrible terms re 4x nd -x.

2 Algebric expressions cn be simplified by using the ssocitive nd distributive properties. -4m(-5n) cn be simplified by rerrnging the terms (we cn do this when the only opertion is multipliction. This is the ssocitive property.) so tht ll the constnts re grouped together nd ll the vribles re grouped together(lphbeticlly)) -4m(-5n) (-4)(-5)mn 0mn (-4z)(y)(-4)()yz -48yz 3(s+7) cn be simplified by using the distributive property 3s + 3(7) 3x + 1 -(-3x - y + 8) cn lso be simplified. Chnge ny subtrction to dding negtive. -(-3x + -y + 8) Now distribute the - nd mke sure to glue the negtive sign on tht - wherever you distribute it! -(-3x) + -(-y) + -(8) 18x + 3y x + 3y Exmple 7: Simplify 4(x - y) (-3x + y) 4x 4y + x - 1y use distributive property to get rid if prentheses 4x + x - 4y -1y regroup terms to put like terms together 10x - 1y combine like terms You do this one: Simplify 7(x - y) 3(-x y)

3 LIKE TERMS Like terms re terms with exctly the sme vribles rised to the exctly the sme powers. Any constnts in n expression re considered like terms. Terms tht re not like terms re clled unlike terms. Like Terms x, 3x, -4x Sme vribles, ech with power of 1. 3, 5, -1 Constnts 5x, -x Sme vribles nd sme powers Unlike Terms x, x Different powers 3, 3x, 3x Different powers 5x, 5y Different vribles Only Like Terms cn be combined! Combining like terms is to dd or subtrct like terms. Combine like terms contining vribles by combining their coefficients nd keeping the sme vribles with the sme exponents. Exmple: 3x 8x (3-8)x -5x Exmple: 5x + x (5 + )x 7x Exmple: 3x + 5x + + x x - 3 3x + x + 5x-x x + 4x 1 Algebric expressions cn only be simplified if they hve like terms, which re terms with the sme vrible nd sme exponent. x Only 5 + cn be simplified. x becomes x + 7 3x + x + cn be simplified to 4x + x + x cnnot be simplified becuse they don t hve the sme exponents nd re therefore not like terms.

4 Algebric expressions cn only be EVALUATED if you re given the vlue for the vribles. Exmple: Evlute x 15 for x 3. Substitute (3) for x in the expression. (3) ( 15) 18 3 Exmple p.8 If nd b-3, evlute b - b Don t forget PEMDAS! ()(-3) - (-3) Do exponents first ()(-3) - 9 Multipliction left to right - 9 Subtrction -15 Exmple 3 p. 8 Evlute when 3 nd b-4 b b [ 3) ( 4) ] [(3) ( 4) ] [ 9 1] [ 7] ( For frctions, ssume tht the numertor nd denomintor re seprte groupings (put [ ] round ech one) nd do wht s inside ech grouping first. Do exponents. Do subtrction in the numertor grouping. Now tht the numertor nd denomintor groupings hve been simplified, you cn divide. You do this one : Evlute for 5 nd b b b

5 Trnslting Verbl Expressions in to Mthemticl Expressions Verbl Expressions Exmples Mth Trnsltion Addition dded to dded to y +y more thn 8 more thn x 8+x the sum of the sum of x nd z x+z incresed by t incresed by 9 t+9 the totl of the totl of 5 nd y 5+y Subtrction minus x minus x- less thn 7 less thn t t-7 subtrcted from 5 subtrcted from decresed by m decresed by 3 m-3 the difference the difference between y nd 4 y-4 between Multipliction times 10 times 10 X of one hlf of (1/) X the product of the product of 4 nd 3 4 X 3 multiplied by y multiplied by 11 11y Division divided by x divided by 1 x/1 the quotient of the quotient of y nd z y/z the rtio of the rtio of t to 9 t/9 Power the squre of the squre of x x the cube of the cube of z z 3 squred y squred y Equivlency equls 1+ equls is is hlf of 4 (½)X4 is the sme s ½ is the sme s /4 1 4 yields 3+1 yields represents y represents x+1 y x + 1 Comprison greter thn -3 is greter thn -5-3 > -5 less thn -5 is less then -3-5 < -3 greter thn or equl x is greter thn or equl to 5 x 5 to t lest x is t lest 80 x 80 no less thn x is no less thn 70 x 70 less thn or equl to x is less then or equl to - x - t most y is t most 3 y 3 no more thn y is no more thn 1 y 1

6 Solving Appliction Problems Problem-Solving Strtegy: Anlyze the problem. Wht re you trying to find? Lbel vribles to the unknown quntities. Wht s the given info? Work out pln before strting. Drw sketch if possible. Look for indictor words (e.g. gined, lost, times, per) to know which opertions (+,-, x, ) to use. Estimte resonble nswer. Solve the problem. Check your work. If the nswer is not resonble, strt over. Exmple on p. 89: The length of swimming pool is 0ft longer thn the width. Express the length of the pool in terms of the width. Step 1) Wht re we trying to find? The length of the pool. Let l length. Wht s the given info? The length is 0ft longer thn the width. Wht s the width? We don t know. So set w width. Step ) Look for indictor words. is mens, longer thn mens + Drw sketch. The length is 0 ft longer thn the width. l 0 + w We hve now expressed the length of the pool in terms of the width. Since we don t know wht the width ctully is, this is s fr s we cn go. The nswer is 0 + w Is this resonble for the length of the pool (this is wht we were sked to express)? Put in ny number of w nd see if the length 0 + w is resonble. Yes. YOU DO THIS ONE: l0+w An older computer tkes twice s long to process set of dt s does new model. Express the mount of time it tkes the older computer to process the dt in terms of the mount of time it tkes the newer model. Let c time older computer tkes to process dt. n time newer computer tkes to process dt. Write c s expression with n in it. w

7 p. 95 #11 A coin bnk contins thirty-five coins in nickels nd dimes. Use the sme vrible to express the number of nickels nd the number of dimes in the coin bnk. Step 1) Wht re we being sked to find? The number of nickels nd dimes expressed with the sme vrible. We ll let d number of dimes. Right now let s use nother vrible, n number of nickels, nd then we ll mke n eqution to express nickels in terms of dimes. Step ) Wht s the given info. Totl coins re 35. Tht is, The number of dimes + the number of nickels 35 d + n 35 Solving for n, we get n 35 d Now we cn sy the number of dimes d nd the number of nickels 35 d Step 3) Is this wht we were sked to find? Yes. You do #10 A hlyrd 1ft long is cut into two pieces. Use the sme vrible to express the lengths of the two pieces. Exmple: p. 9 #17 A wire whose length is given s x inches is bent into squre. Express the length of side of the squre in terms of x. Drw picture. We re sked to express the length of side of the squre in terms of x. Let s let s length of side of the squre. Wht s given? The wire (of length x) is bent into squre. x Wht do we know bout squres? The perimeter (or length going round the squre) is s + s + s +s, nd the totl length going round the squre is x (given). s + s+ s+s x 1s + 1s +1s + 1s x 4s x s x/4