Answer:
Matrix :
![\begin{bmatrix}1&1&1&|&380\\ 5&3&10&|&1460\\ -2&1&0&|&0\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D1%261%261%26%7C%26380%5C%5C%205%263%2610%26%7C%261460%5C%5C%20-2%261%260%26%7C%260%5Cend%7Bbmatrix%7D)
Solution Set : { x = 123, y = 246, z = 11 }
Step-by-step explanation:
Let's say that x represents the number of car wash tickets, y represents the number of silly sting fight tickets, and z represents the number of dance tickets. We know that the total tickets = 380, so therefore,
x + y + z = 380,
And the car wash tickets were $5 each, the silly sting fight tickets were $3 each and the dance tickets were $10 each, the total cost being $1460.
5x + 3y + 10z = 1460
The silly string tickets were sold for twice as much as the car wash tickets.
y = 2x
Therefore, if we allign the co - efficients of the following system of equations, we get it's respective matrix.
System of Equations :
![\begin{bmatrix}x+y+z=380\\ 5x+3y+10z=1460\\ y=2x\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7Dx%2By%2Bz%3D380%5C%5C%205x%2B3y%2B10z%3D1460%5C%5C%20y%3D2x%5Cend%7Bbmatrix%7D)
Matrix :
![\begin{bmatrix}1&1&1&|&380\\ 5&3&10&|&1460\\ -2&1&0&|&0\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D1%261%261%26%7C%26380%5C%5C%205%263%2610%26%7C%261460%5C%5C%20-2%261%260%26%7C%260%5Cend%7Bbmatrix%7D)
Let's reduce this matrix to row - echelon form, receiving the number of car wash tickets, silly sting fight tickets, and dance tickets,
- Swap Matrix Rows
- Cancel leading Co - efficient in second row
- Cancel leading Co - efficient in third row
- Swap second and third rows
- Cancel leading co - efficient in row three
And we can continue, canceling the leading co - efficient in each row until this matrix remains,
![\begin{bmatrix}1&0&0&|&\frac{2340}{19}\\ 0&1&0&|&\frac{4680}{19}\\ 0&0&1&|&\frac{200}{19}\end{bmatrix}](https://tex.z-dn.net/?f=%5Cbegin%7Bbmatrix%7D1%260%260%26%7C%26%5Cfrac%7B2340%7D%7B19%7D%5C%5C%200%261%260%26%7C%26%5Cfrac%7B4680%7D%7B19%7D%5C%5C%200%260%261%26%7C%26%5Cfrac%7B200%7D%7B19%7D%5Cend%7Bbmatrix%7D)
x = 2340 / 19 = ( About ) 123 car wash tickets sold, y= 4680 / 19 =( About ) 246 silly string fight tickets sold, z = 200 / 19 = ( About ) 11 tickets sold