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A linear integral question.?

Stephen decides to spend up $300 to buy x pairs of shoes,

y shirts & z jeans. A pair of shoes costs $40; a jacket $68

& a jean $19. How many of each of the items can he buy?

Help please.

3 Answers

Relevance
  • 3 months ago

    (My way)

    40x+68y+19z=300=>

    z=(300-40x-68y)/19=>

    z=15-2x-3y+[15-(2x+11y)]/19=>

    z=15-2x-3y+[15-t]/19, where t=2x+11y,=>

    z=15-2x-3y-k1 {(15-t)/19 is an integer=>t=15(mod 19)=>

    .........................t=15+19k1, k1 is an integer}

    15+19k1=2x+11y=>

    y=1+k1+[4+(8k1-2x)]/11=>

    y=1+k1+(4+s)/11, where s=8k1-2x,=>

    y=2+k1+k2 {(4+s)/11 is an integer=>s=7(mod 11)=>

    ....................s=7+11k2, k2 is an integer}

    2x=8k1-s=>

    x=4k1-(7+11k2)/2=>

    x=4k1-9-11k3 {(7+11k2)/2 is an integer=>k2=1(mod 2)=>

    ........................k2=1+2k3, k3 is an integer}

    y=2+k1+k2=>

    y=3+k1+2k3=>

    z=24-12k1+16k3

    Thus, the G.S. is

    x=4k1-11k3-9>=0-------(1)

    y=k1+2k3+3>=0---------(2)

    z=-12k1+16k3+24>=0------(3)

    In the diagram below, the feasible integral

    solution is found to be k1=0, k3=-1 inside

    the region ABC which is formed by the

    boundary lines of (1), (2) & (3).

    So, x=2, y=1 & z=8 is the only choice

    for Stephen.

    Attachment image
  • I'm going to assume that jacket and shirt are interchangeable here.

    40x + 68y + 19z = 300

    68y + 40x + 19z = 300

    4 < 300/68 < 5

    4 * 68 = 240 + 32 = 272

    3 * 68 = 204

    2 * 68 = 136

    1 * 68 = 68

    0 * 68 = 0

    (shirts, shoes, jeans , change)

    (0 , 0 , 0 , 300)

    (0 , 0 , 1 , 281)

    (0 , 0 , 2 , 262)

    (0 , 0 , 3 , 243)

    (0 , 0 , 4 , 224)

    (0 , 0 , 5 , 205)

    (0 , 0,  6 , 186)

    (0 , 0 , 7 , 167)

    (0 , 0 , 8 , 148)

    (0 , 0 , 9 , 129)

    (0 , 0 , 10 , 110)

    (0 , 0 , 11 , 91)

    (0 , 0 , 12 , 72)

    (0 , 0 , 13 , 53)

    (0 , 0 , 14 , 34)

    (0 , 0 , 15 , 15)

    (0 , 1 , 0 , 260)

    (0 , 1 , 1 , 241)

    (0 , 1 , 2 , 222)

    (0 , 1 , 3 , 203)

    (0 , 1 , 4 , 184)

    (0 , 1 , 5 , 165)

    (0 , 1 , 6 , 146)

    (0 , 1 , 7 , 127)

    (0 , 1 , 8 , 108)

    (0 , 1 , 9 , 89)

    (0 , 1 , 10 , 70)

    (0 , 1 , 11 , 51)

    (0 , 1 , 12 , 32)

    (0 , 1 , 13 , 13)

    (0 , 2 , 0 , 220)

    (0 , 2 , 1 , 201)

    (0 , 2 , 2 , 182)

    (0 , 2 , 3 , 163)

    (0 , 2 , 4 , 144)

    (0 , 2 , 5 , 125)

    (0 , 2 , 6 , 106)

    (0 , 2 , 7 , 87)

    (0 , 2 , 8 , 68)

    (0 , 2 , 9 , 49)

    (0 , 2 , 10 , 30)

    (0 , 2 , 11 , 11)

    (0 , 3 , 0 , 180)

    (0 , 3 , 1 , 161)

    (0 , 3 , 2 , 142)

    (0 , 3 , 3 , 123)

    (0 , 3 , 4 , 104)

    (0 , 3 , 5 , 85)

    (0 , 3 , 6 , 66)

    (0 , 3 , 7 , 47)

    (0 , 3 , 8 , 28)

    (0 , 3 , 9 , 9)

    (0 , 4 , 0 , 140)

    (0 , 4 , 1 , 121)

    (0 , 4 , 2 , 102)

    (0 , 4 , 3 , 83)

    (0 , 4 , 4 , 64)

    (0 , 4 , 5 , 45)

    (0 , 4 , 6 , 26)

    (0 , 4 , 7 , 7)

    (0 , 5 , 0 , 100)

    (0 , 5 , 1 , 81)

    (0 , 5 , 2 , 62)

    (0 , 5 , 3 , 43)

    (0 , 5 , 4 , 24)

    (0 , 5 , 5 , 5)

    (0 , 6 , 0 , 60)

    (0 , 6 , 1 , 41)

    (0 , 6 , 2 , 22)

    (0 , 6 , 3 , 3)

    (0 , 7 , 0 , 20)

    (0 , 7 , 1 , 1)

    (1 , 0 , 0 , 332)

    (1 , 0 , 1 , 313)

    (1 , 0 , 2 , 294)

    (1 , 0 , 3 , 275)

    (1 , 0 , 4 , 256)

    (1 , 0 , 5 , 237)

    (1 , 0 , 6 , 218)

    (1 , 0 , 7 , 199)

    (1 , 0 , 8 , 180)

    (1 , 0 , 9 , 161)

    (1 , 0 , 10 , 142)

    (1 , 0 , 11 , 123)

    (1 , 0 , 12 , 104)

    (1 , 0 , 13 , 85)

    (1 , 0 , 14 , 66)

    (1 , 0 , 15 , 47)

    (1 , 0 , 16 , 28)

    (1 , 0 , 17 , 9)

    (1 , 1 , 0 , 292)

    (1 , 1 , 1 , 273)

    (1 , 1 , 2 , 254)

    (1 , 1 , 3 , 235)

    (1 , 1 , 4 , 216)

    (1 , 1 , 5 , 197)

    (1 , 1 , 6 , 178)

    (1 , 1 , 7 , 159)

    (1 , 1 , 8 , 140)

    (1 , 1 , 9 , 121)

    (1 , 1 , 10 , 102)

    (1 , 1 , 11 , 83)

    (1 , 1 , 12 , 64)

    (1 , 1 , 13 , 45)

    (1 , 1 , 14 , 26)

    (1 , 1 , 15 , 7)

    (1 , 2 , 0 , 252)

    (1 , 2 , 1 , 233)

    (1 , 2 , 2 , 214)

    (1 , 2 , 3 , 195)

    (1 , 2 , 4 , 176)

    (1 , 2 , 5 , 157)

    (1 , 2 , 6 , 138)

    (1 , 2 , 7 , 119)

    (1 , 2 , 8 , 100)

    (1 , 2 , 9 , 81)

    (1 , 2 , 10 , 62)

    (1 , 2 , 11 , 43)

    (1 , 2 , 12 , 24)

    (1 , 2 , 13 , 5)

    (1 , 3 , 0 , 212)

    (1 , 3 , 1 , 193)

    (1 , 3 , 2 , 174)

    (1 , 3 , 3 , 155)

    (1 , 3 , 4 , 136)

    (1 , 3 , 5 , 117)

    (1 , 3 , 6 , 98)

    (1 , 3 , 7 , 79)

    (1 , 3 , 8 , 60)

    (1 , 3 , 9 , 41)

    (1 , 3 , 10 , 22)

    (1 , 3 , 11 , 3)

    (1 , 4 , 0 , 172)

    (1 , 4 , 1 , 153)

    (1 , 4 , 2 , 134)

    (1 , 4 , 3 , 115)

    (1 , 4 , 4 , 96)

    (1 , 4 , 5 , 77)

    (1 , 4 , 6 , 58)

    (1 , 4 , 7 , 39)

    (1 , 4 , 8 , 20)

    (1 , 4 , 9 , 1)

    (1 , 5 , 0 , 132)

    (1 , 5 , 1 , 113)

    (1 , 5 , 2 , 94)

    (1 , 5 , 3 , 75)

    (1 , 5 , 4 , 56)

    (1 , 5 , 5 , 37)

    (1 , 5 , 6 , 18)

    (1 , 6 , 0 , 92)

    (1 , 6 , 1 , 73)

    (1 , 6 , 2 , 54)

    (1 , 6 , 3 , 35)

    (1 , 6 , 4 , 16)

    (1 , 7 , 0 , 92)

    (1 , 7 , 1 , 73)

    (1 , 7 , 2 , 54)

    (1 , 7 , 3 , 35)

    (1 , 7 , 4 , 16)

    (1 , 8 , 0 , 52)

    (1 , 8 , 1 , 33)

    (1 , 8 , 2 , 14)

    (1 , 9 , 0 , 12)

    (2 , 0 , 0 , 164)

    (2 , 0 , 0 , 145)

    (2 , 0 , 0 , 126)

    (2 , 0 , 0 , 107)

    (2 , 0 , 0 , 88)

    (2 , 0 , 0 , 69)

    (2 , 0 , 0 , 50)

    (2 , 0 , 0 , 31)

    (2 , 0 , 0 , 12)

    (2 , 1 , 0 , 124)

    (2 , 1 , 1 , 105)

    (2 , 1 , 2 , 86)

    (2 , 1 , 3 , 67)

    (2 , 1 , 4 , 48)

    (2 , 1 , 5 , 29)

    (2 , 1 , 6 , 10)

    (2 , 2 , 0 , 84)

    (2 , 2 , 1 , 65)

    (2 , 2 , 2 , 46)

    (2 , 2 , 3 , 27)

    (2 , 2 , 4 , 8)

    (2 , 3 , 0 , 44)

    (2 , 3 , 1 , 25)

    (2 , 3 , 2 , 6)

    (3 , 0 , 0 , 96)

    (3 , 0 , 1 , 77)

    (3 , 0 , 2 , 58)

    (3 , 0 , 3 , 39)

    (3 , 0 , 4 , 20)

    (3 , 0 , 5 , 1)

    (3 , 1 , 0 , 56)

    (3 , 1 , 1 , 37)

    (3 , 1 , 2 , 18)

    (3 , 2 , 0 , 16)

    (4 , 0 , 0 , 28)

    (4 , 0 , 1 , 9)

    Looks like 199 distinct solutions to me.

  • 3 months ago

    Stephen decides to spend up $300 to buy x pairs of shoes, y shirts & z jeans. 

    A pair of shoes costs $40; a jacket $68 & a jean $19. 

    How many of each of the items can he buy?

    40x + 68y + 19z = 300                                                

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