Kinda irritating math problem

person spent $14.00 on postage staps
the stamps were $0.15, $0.20, and $0.35
they bought 45 stamps

use the gauss-jordan elimination method (matrices) to solve and show all work

X=number of stamps at a price of $0.15
y=number of stamps at a price of $0.20
z=number of stamps at a price of $0.35

We have a system of two equations with three unknowns

x+y+z=45
0.15x+0.20y+0.35 z=14

We solve this system of equations by Gauss-Jordan elimination method.

1             1           1             45
0.15      0.2        0.35         14

1             1           1             45
15           20         35          1400        (100R₂)

1             1           1             45
0             5           20           725          (R₂-15R₁)

1             1           1             45
0             1           4           145         (R₂/5)    

Z=λ;        λ∈N;     λ∈[0,45]
y+4λ=145      ⇒    y=145-4λ

x+(145-4λ)+λ=45
x=45-λ+4λ-145
x=-100+3λ

The solution is:
x=-100+3λ
y=145-4λ
z=λ                        λ∈[0.45]

Now, we calculate the possible solutions:
x=-100+3λ
x≥0
-100+3λ≥0
λ=≥33.33333⇒λ can be: 34,35,36,45;

y=145-4λ
y≥0
145-4λ≥0
-4λ≥-145
-λ≥-145/4
λ≤36.25        ⇒λ can be (34,35 or 36)

if λ=34
z=34
y=145-4λ=145-136=9
x=-100+3 λ=-100+102=2

if λ=35
z=35
y=145-4λ=145-140=5
x=-100+3λ=-100+105=5

if λ=36
z=36
y=145-4λ=145-144=1
x=-100+3λ=-100+108=8

We have threes possible solutions:
solution 1:
2 stamps at a price of $0.15
9 stamps at a price of $0.2
34 stamps at a price of $0.35

 solution 2:
5 stamps at a price of $0.15
5 stamps at a price of $0.2
35 stamps at a price of $0.35

solution 3:
8 stamps at a price of $0.15
1 stamps at a price of $0.2
36 stamps at a price of $0.35

Solutions 

To solve this problem lets use the variables >> "x,y,and z"

x=number of stamps at a price of $0.15
y=number of stamps at a price of $0.20
z=number of stamps at a price of $0.35

We have a system of two equations as the three unknowns

x+y+z=45

0.15x+0.20y+0.35 z=14

We solve this system of equations by Gauss-Jordan elimination method.The Gauss-Jordan elimination method is used to solve a system of linear equations.  



 

 (100R₂) 

 

(R₂-15R₁)
 


 (R₂/5)     

Calculations 

Z=λ;        λ∈N;     λ∈[0,45] 
y+4λ=145      ⇒    y=145-4λ

x+(145-4λ)+λ=45
x=45-λ+4λ-145
x=-100+3λ

The solution is:
x=-100+3λ
y=145-4λ
z=λ                      ≡  λ∈[0.45] 

Our next step is to calculate all the possible solutions. 

x=-100+3λ
x≥0
-100+3λ≥0
λ=≥33.33333....⇒λ can be numbers: 34,35,36,....45;

y=145-4λ
y≥0
145-4λ≥0
-4λ≥-145
-λ≥-145/4
λ≤36.25        ⇒λ can be the numbers (34,35 or 36)

if λ=34
z=34
y=145-4λ=145-136=9
x=-100+3 λ=-100+102=2

if λ=35
z=35
y=145-4λ=145-140=5
x=-100+3λ=-100+105=5

if λ=36
z=36
y=145-4λ=145-144=1
x=-100+3λ=-100+108=8 

Simplify 

We have three solutions 

⇒solution 1:

2 stamps at a price of $0.15
9 stamps at a price of $0.2
34 stamps at a price of $0.35

 ⇒solution 2:

5 stamps at a price of $0.15
5 stamps at a price of $0.2
35 stamps at a price of $0.35 

⇒solution 3:

8 stamps at a price of $0.15
1 stamps at a price of $0.2
36 stamps at a price of $0.35  

Note - λ∈N = λ belong to natural numbers (1,2,3,4....)

Where are the