how to solve a word problem using Matrices

Question

Philip made a total of 9 bracelets and necklaces from 120 inches of cord.
He used 8 inches of cord for each bracelet and 20 inches of cord for each necklace.
Use matrices to find the number of bracelets and the number of necklaces that Philip made.

Answer 1

B=number of bracelets, N=number of necklaces.

B+N=9 (total number)

8B+20N=120 (total length of cord).

Matrix is:

⎛ 1   1 ⎞

⎝ 8 20 ⎠

Determinant Δ is 1×20-8×1=12.

Replace the first column with the numbers (total and cord length):

⎛ 9       1 ⎞

⎝ 120 20 ⎠

B-determinant ΔB=9×20-120×1=60, B=ΔB/Δ=60/12=5.

Replace the second column with the same numbers:

⎛ 1     9 ⎞

⎝ 8 120 ⎠

N-determinant ΔN=120-72=48, N=ΔN/Δ=48/12=4.

CHECK

B+N=5+4=9, 8B+20N=40+80=120 inches. This confirms 5 bracelets and 4 necklaces.