Final answer:
To write the system of linear equations in augmented matrix form, arrange the coefficients and constant terms. Use Gaussian elimination to solve the system by performing row operations on the augmented matrix. The concentrations in the three vats are all zero.
Step-by-step explanation:
To write the given system of linear equations in augmented matrix form, we can arrange the coefficients of the variables and the constant terms in a rectangular array. Each row of the array represents an equation in the system, and the last column represents the constant term in each equation. The augmented matrix for the given system of equations is:
[[-2, 7, -6, 20],[-8, 4, -1, -43],[6, -17, 13, -43]]
To solve the system using Gaussian elimination, we perform row operations on the augmented matrix to transform it into row-echelon form or reduced row-echelon form. We can use row operations such as multiplying a row by a scalar, adding a multiple of one row to another row, or swapping two rows.
Here are the steps:
1. Multiply the first row by -4 and add it to the second row: [0, -3, 23, 13]
2. Multiply the first row by 3 and add it to the third row: [0, -61, 31, -23]
3. Multiply the second row by -20 and add it to the third row: [0, -61, 31, -23]
4. Divide the second row by -61: [0, 1, -31/61, 23/61]
5. Multiply the third row by -13 and add it to the second row: [0, 1, 0, 0]
6. Multiply the second row by 23/61: [0, 23/61, 0, 0]
7. Multiply the second row by 7 and subtract it from the first row: [1, 0, 0, 0]
So, the row-echelon form of the augmented matrix is:
[[1, 0, 0, 0],[0, 1, 0, 0],[0, 0, 1, 0]]
From the row-echelon form, we can see that C₁ = 0, C₂ = 0, and C₃ = 0. Therefore, the concentrations in the three vats are all zero.