Question

A farmer's market sells pecans, walnuts, and cashews. They each have a different cost and are priced per pound. Alyssa buys five pounds of walnuts and seven pounds of cashews and spends $75. Landon buys two pounds of pecans and three pounds of walnuts and spends $31. Tina buys 4 pounds of pecans and 3 pounds of cashews and spends $29. Find the missing quantity below.

Price per pound for Walnuts = $_______________

Answer 1

Price per pound for Walnut is $8

Explanation:

Represent Walnuts with W, Pecans with P and Cashew with C

For Alyssa:

`5W + 7C = 75`

For Landon:

`2P + 3W = 31`

For Tina:

`4P + 3C = 29`

Required

Solve for W

Considering the last equation, make P the subject:

`4P + 3C = 29`

`4P = 29 - 3C`

`P = (1)/(4)(29 - 3C)`

Substitute this expression for P in the second equation

`2P + 3W = 31`

`2((1)/(4)(29 - 3C)) + 3W = 31`

`(1)/(2)(29 - 3C) + 3W = 31`

`(29)/(2) - (3C)/(2) + 3W = 31`

`3W - (3C)/(2)= 31 - (29)/(2)`

Multiply through by 2

`6W - 3C = 62 - 29`

`6W - 3C = 33`

Make C the subject

`-3C = 33 - 6W`

`C = (33)/(-3) - (6W)/(-3)`

`C = -11 + 2W`

`C = 2W - 11`

Substitute this in the first equation

`5W + 7C = 75`

`5W + 7(2W - 11) = 75`

`5W + 14W - 77 = 75`

`5W + 14W = 75 + 77`

`19W = 152`

`W = 152/19`

`W = 8`

Price per pound for Walnut is $8

Answer 2

price per pound of walnut = x = $8

price per pound of pecans = y = $3.5

price per pound of cashew = z = $5

Explanation:

Let

price per pound of walnut = x price per pound of pecans = y price per pound of cashew = z.

Alyssa;

5x + 7z = 75

Landon:

2y + 3x = 31

Tina:

4y + 3z = 29

5x + 7z = 75 (1)

2y + 3x = 31 (2)

4y + 3z = 29 (3)

From (1)

5x + 7z = 75

5x = 75 - 7z

x = (75 - 7z) / 5

Substitute x = (75 - 7z) / 5 into (2)

2y + 3x = 31

2y + 3((75 - 7z) / 5 = 31

2y + (225 - 21z) /5 = 31

(225 - 21z) /5 = 31 - 2y

225 - 21z = 5(31 - 2y)

225 - 21z = 155 - 10y

225 - 155 = -10y + 21z

70 = -10y + 21z

-10y + 21z = 70

4y + 3z = 29 (3)

-10y + 21z = 70 (4)

Multiply (3) by 7

28y + 21z = 203. (3b)

-10y + 21z = 70 (4)

Subtract (4) from (3b)

28y - (-10y) = 203 - 70

28y + 10y = 133

38y = 133

y = 133/38

= 3.5

y = 3.5

Substitute y = 3.5 into (3)

4y + 3z = 29 (3)

4(3.5) + 3z = 29

14 + 3z = 29

3z = 29 - 14

3z = 15

z = 15/3

z = 5

Substitute z = 5 into (1)

5x + 7z = 75 (1)

5x + 7(5) = 75

5x + 35 = 75

5x = 75 - 35

5x = 40

x = 40/5

x = 8

price per pound of walnut = x = $8

price per pound of pecans = y = $3.5

price per pound of cashew = z = $5