for the following problem (Matrices)

Question

represent the system of equations with a matrix, then convert to an upper triangular form and solve for variables with back substitution: A can is filled with 45 coins: half-dollars, quarters, and dimes. There are four times as many half-dollars as quarters. The total value of all the coins is $15. How many of each coin are there?

Answer 1

H = 4Q ---> h - 4Q = 0

H + Q + D = 45

50H + 25Q + 10D = 1500 <--- amounts in pennies to clear away the decimals

h         q           d              k
-----------------------------------
1         -4          0              0
1          1          1              45
50       25        10             1500


h            q            d            k
-----------------------------------
1           -4            0            0
0            5            1           45
0           225        10         1500

-R1 + R2; -50*R1 + R3


h       q             d             k
-----------------------------------
1      -4             0             0
0       1            1/5           9
0      225           10          1500

r2/5


h       q        d           k
-----------------------------------
1       0       4/5         36
0       1       1/5          9
0       0       -35        -525


4*R2 + R1; -225*R2 + R3



h    q       d          k
-----------------------------------
1    0     4/5         36
0    1     1/5           9
0    0      1            15

R3/-35


Backwards Substitution:

d = 15

q + 1/5d = 9
q + 1/5(15) = 9
q + 3 = 9
q = 6

h + 4/5d = 36
h + (4/5)(15) = 36
h + 12 = 26
h = 24

------ Here's the algebraic solution ----------------------

H = 4Q

H + Q + D = 45

50H + 25Q + 10D = 1500 <--- amounts in pennies to clear away the decimals

------- substitutes first equation into second and third -------------------------
4Q + Q + D = 45

50(4Q) + 25Q + 10D = 1500


--------------- simplifies ----------------



5Q + D = 45

225Q + 10D = 1500

------------------------


225Q + 10 (45 - 5Q) = 1500 <---- solves for D in top equation and plugs into bottom equation

225q + 450 - 50q = 1500 <--- distributive

175q = 1050 <---- combines like terms, then subtracts 450 from both sides


Q= 1050/175 = 6

d =45 - 5q = 45 - 5(6) = 45 - 30 = 15

h = 4q = 4(6) = 24

CHECK 6*25 + 15*10 + 24*50 = 150 + 150 + 1200 = 1500