systems of equation

Question

A baker makes three types of cookies: Chocolate Chip, Snickernoodles, and Peanut Butter cookies. A batch of Chocholate Chip cookies requires 2 eggs, 3 cups of flour, and 1 1/2 cups of sugar. A batch of Snickernoodles requires 2 eggs, 2 1/2 cups of flour, and 2 cups of sugar. A batch of Peanut Butter cookies requires 1 egg, 2 3/4 cups of flour, and 2 1/2 cups of sugar. The baker has a dozen eggs, however, one egg is broken. He has 16 1/4 cups of flour, and 11 1/2 cups of sugar. How many batches of each type of cookie can the baker make to use all available ingredients?

Answer 1

CC=2e+3f+1.5s where e=eggs, f=flour, and s=sugar

SN=2e+2.5f+2s

PB=1e+2.75f+2.5s

x=# of CC batches, y=# of SN batches, and z=# of PB batches

 

2ex+2ey+ez=11e

3fx+2.5fy+2.75fz=16.25f

1.5sx+2sy+2.5sz=11.5s

 

e(2x+2y+z)=11e

f(3x+2.5y+2.75z)=16.25f

s(1.5x+2y+2.5z)=11.5s

 

2x+2y+z=11

3x+2.5y+2.75z=16.25

1.5x+2y+2.5z=11.5

2(1.5x+2y+2.5z)=2(11.5) gives us 3x+4y+5z=23

subtract the second equation from the new third equation

1.5y+2.25z=6.75

 

take the first two equations:

3(2x+2y+z)=3(11)

2(3x+2.5y+2.75z)=2(16.25)

multiplying we get...

6x+6y+3z=33

6x+5y+5.5z=32.5

subtract the equations:

y-2.5z=0.5

y=2.5z+0.5

use 1.5y+2.25z=6.75 and substitute 2.5z+0.5 for y

1.5(2.5z+0.5)+2.25z=6.75

3.75z+0.75+2.25z=6.75

6z+0.75=6.75

6z=6

z=1

y-2.5z=0.5

y-2.5=0.5

y=3

2x+2y+z=11

2x+2(3)+1=11

2x+6+1=11

2x+7=11

2x=4

x=2

You can bake 2 batches of CC, 3 batches of SN, and 1 batch of PB

Answer 2

eggs = 11 = 2 C + 2 S + 1 P
flour = 16 1/4 = 3 C + 2 1/2 S + 2 3/4 P
sugar = 11 1/2 = 1 1/2 C + 2 S + 2 1/2 P

Input to GeoGebra:

S = ReducedRowEchelonForm[{{2,2,1,11},{3,2.5,2.75,16.25},{1.5,2,2.5,11.5}}]
=>
2 batches of Chocolate Chip cookies,
3 batches of Snickernoodles, and
1 batch of Peanut Butter cookies.

==========

Form Augmented Matrix and perform Row Operations to get an Identity Matrix in the first 3 columns; then solution vector will be in 4th column.

R1 is row 1, etc.

R1 = {2,2,1,11}
R2 = {3,2.5,2.75,16.25}
R3 = {1.5,2,2.5,11.5}

R2 >> R2*4
R3 >> R3*2

R1 = {2,2,1,11}
R2 = {12,10,11,65}
R3 = {3,4,5,23}

R1 >> 6*R1
R2 >> -R2
R3 >> -4*R3

R1 = {12,12,6,66}
R2 = {-12,-10,-11,-65}
R3 = {-12,-16,-20,-92}

R2 >> R2+R1
R3 >> R3+R1

R1 = {12,12,6,66}
R2 = {0,2,-5,1}
R3 = {0,-4,-14,-26}

R1 >> R1/6
R2 >> 2*R2

R1 = {2,2,1,11}
R2 = {0,4,-10,2}
R3 = {0,-4,-14,-26}

R3 >> R3+R2

R1 = {2,2,1,11}
R2 = {0,4,-10,2}
R3 = {0,0,-24,-24}

R2 >> R2/2
R3 >> R3/(-24)

R1 = {2,2,1,11}
R2 = {0,2,-5,1}
R3 = {0,0,1,1}

R1 >> R1-R3
R2 >> R2+5*R3

R1 = {2,2,0,10}
R2 = {0,2,0,6}
R3 = {0,0,1,1}

R1 >> R1-R2

R1 = {2,0,0,4}
R2 = {0,2,0,6}
R3 = {0,0,1,1}

R1 >> R1/2
R2 >> R2/2

R1 = {1,0,0,2}
R2 = {0,1,0,3}
R3 = {0,0,1,1}

 

This is a Reduced Row Echelon Form matrix with the solution in the 4th column.

 

Solution is:

2 batches of Chocolate Chip cookies,
3 batches of Snickernoodles, and
1 batch of Peanut Butter cookies.