Final answer:
An augmented matrix for the system with the solution x=-16+10t, y=-15+10t, z=2-2t is | 10 -1 0 | 16 |, | 10 0 -1 | 15 |, | -2 0 -1 | -2 |. The matrix's rows and columns correspond to equations and variables, respectively.
Step-by-step explanation:
To create an augmented matrix representing a system of linear equations with the solution x = -16 + 10t, y = -15 + 10t, z = 2 - 2t, we need to construct a system where each row represents a linear equation, and each column represents a coefficient of a variable, with the last column after the augmentation representing the constant terms. We can set up the coefficients of t as the coefficients of a 'parameter variable'. Here's how we can form the equations:
- Equation 1 (for x): 10t - x = 16
- Equation 2 (for y): 10t - y = 15
- Equation 3 (for z): -2t - z = -2
Now, writing the augmented matrix, we get:
| 10 -1 0 | 16 |
| 10 0 -1 | 15 |
| -2 0 -1 | -2 |
Each row of the matrix corresponds to an equation, and each column corresponds to one of the variables or the constant term (last column).