Final answer:
The linear system x + y - z + 2w = 4, w = 5 can be solved by substituting w = 5 into the first equation, resulting in x + y - z = -6. This system is underdetermined with infinitely many solutions. A particular solution can be given by setting two variables to arbitrary values and solving for the third.
Step-by-step explanation:
To solve the linear system given in row echelon form, we have the equations x + y - z + 2w = 4, and w = 5. Since we already know the value of w, we can substitute it directly into the first equation:
- x + y - z + 2(5) = 4
- x + y - z + 10 = 4
- x + y - z = -6
Now we are left with a system of two equations and three unknowns (x, y, z). This is underdetermined, meaning there are an infinite number of solutions depending on the values chosen for x and y. To find particular solutions, you can set two of the variables to arbitrary values and solve for the third. For example, you could set x = 0 and solve for y and z, or set y = 0 and solve for x and z, and so on.
The final step involves expressing the solutions in terms of the free variable(s). For instance, if we set x = 0, then we would have:
In this case, we could express z in terms of y: z = y + 6.
Hence, the solution set can be described by two parameters, y and z, where z is dependent on y.