Let d be the cost to rent one documentary, v be the cost to rent one video game, and m be the cost to rent one movie. In the first week Collen rented 3 documentaries, 2 video games, and 2 movies and the charge was $16.29. Therefore, we can write the first equation.
3d+ 2v+ 2m=16.29 (I)In the next week, she rented 1 documentary, 3 video games, and 4 movies for a total charge of $19.84. Therefore, we can write the second equation.
1d+ 3v+ 4m=19.84 ⇔ d+3v+4m=19.84 (II)
The third week she rented 2 documentaries, 1 video game, and 1 movie for a total charge of $9.14. Therefore, we can write the last equation.
2d+ 1v+ 1m=9.14 ⇔ 2d+v+m=9.14 (III)
Combining Equations I, II, and III together, we get the system of equations we are looking for.
3d+2v+2m=16.29 & (I) d+3v+4m=19.84 & (II) 2d+v+m=9.14 & (III)
3d+ 2v+ 2m=16.29 1d+ 3v+ 4m=19.84 2d+ 1v+ 1m=9.14
Let C be the coefficient matrix of the system. The coefficient matrix is a matrix that contains only the coefficients of a system.
C=
[
ccc
3 & 2 & 2
1 & 3 & 4
2 & 1 & 1
]
If the determinant of the coefficient matrix, |C|, is different from zero, we can find the solution to our system, (d,v,m), by using determinants. If you need clarification on how to obtain the formulas below, please see the explanation at the bottom of the solution.
d=| cc 16.29 & 2 & 2 19.84 & 3 & 4 9.14 & 1 & 1 |/|C|
v=| cc 3 & 16.29 & 2 1 & 19.84 & 4 2 & 9.14 & 1 |/|C|
m=| cc 3 & 2 & 16.29 1 & 3 & 19.84 2 & 1 & 9.14 |/|C|
Let's start by calculating |C|.
[ 3* 3* 1+ 2* 4* 2+ 2* 1* 1]
-
[ 2* 3* 2+ 1* 4* 3+ 1* 1* 2]
⇕
27-26=1
Let's substitute 1 for |C| in the corresponding formula to find the value of d.
The solution of the system is d=1.99, v=2.79, and m=2.37. Therefore, the cost to rent one documentary is $1.99, one video game is $2.79, and one movie is $2.79.
Extra
Obtaining the formulas for d, v, and m.
Consider our system.
3d+ 2v+ 2m=16.29 1d+ 3v+ 4m=19.84 2d+ 1v+ 1m=9.14
To find the value of the d-variable, we need to calculate the quotient between two determinants. As we have seen before, the divisor will be |C|, the determinant of the coefficient matrix. To find the dividend, substitute in C the result of each of the three equations for the coefficients of d. Then, calculate the determinant of the resulting matrix.
result column
↓
d=| cc 16.29 & 2 & 2 19.84 & 3 & 4 9.14 & 1 & 1 |/|C|
We find v and m in a similar way. In the coefficient matrix, substitute the results of the equations for the respective coefficients. Calculate the determinant of the resulting matrices and divide by the determinant of the coefficient matrix, |C|.
result column result column
↓ ↓
v=| cc 3 & 16.29 & 2 1 & 19.84 & 4 2 & 9.14 & 1 |/|C|
m=| cc 3 & 2 & 16.29 1 & 3 & 19.84 2 & 1 & 9.14 |/|C|
