Suppose that a fourth equation −2w + y = 5 is included in the system (∗). What is
the solution of the resulting system? Answer: ( , , , )

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We are told there is a system of equations that now includes a fourth equation: -2w + y = 5. To reason about the given answer choices, we can use this relation to constrain w and y, since y must equal 5 + 2w.

First, consider Option 1: (w, x, y, z) = (2, 3, t, -1).
- Here w = 2, so from -2w + y = 5 we get y = 5 + 2w = 5 + 4 = 9. That means t must be 9 for this option to satisfy the fourth equation. If t is not 9, this option is inconsistent with the fourth equation. We also note z = -1 but the fourth equation does not involve z, so z being -1 is not constrained by this equation alone.

Next, Option 2: (w, x, y, z) = (3, 2, t, -1).
- With w = 3, the relation gives y = 5 + 2w = 5 + 6 = 11. Therefore t must be 11 for this option to satisfy -2w + y = 5. As with Option 1, z = -1 is not constrained by the fourth equation.

Option 3: (w, x, y, z) = (4, 3, 2, 1).
- Here w = 4 and y = 2. Plugging into -2w + y = 5 gives -2(4) + 2 = -8 + 2 = -6, which does not equal 5. This option directly violates the fourth equation, so it cannot be a valid solution to the augmented system regardless of the other equations.

Option 4: (w, x, y, z) = (2, 2, t, -2).
- With w = 2, the required y is y = 5 + 2w = 9, so t must be 9. Here z = -2; again, the fourth equation has no bearing on z, so z can be -2 as long as the other equations allow it. This option would be consistent with the fourth equation if t = 9.

Summarizing the checks:
- Option 3 is immediately inconsistent with -2w + y = 5, so it cannot be a solution.
- Options 1 and 2 could be consistent only if t matches the computed value from y = 5 + 2w (t = 9 for Option 1 with w = 2, and t = 11 for Option 2 with w = 3).
- Option 4 could be consistent if t = 9, given w = 2.

Because the actual value of t depends on the other equations in the system (which determine w, y, and thus t), we cannot declare definitively which option is correct without those original equations. However, we can clearly see which options are ruled out by the fourth equation alone (Option 3), and for the remaining options, we know the necessary t values that would satisfy -2w + y = 5:
- Option 1 requires t = 9.
- Option 2 requires t = 11.
- Option 4 requires t = 9.

Keep these t-value requirements in mind when you reference the rest of the system to determine which option(s) could be valid in context.

Solomon Systems of Linear Equations - Quiz 1

Suppose that a fourth equation −2w + y = 5 is included in the system (∗). What is the solution of the resulting system? Answer: ( , , , )

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