https://file.aiquickdraw.com/mathbot/file/172891764266316youa6w.jpg

Let's begin with the first part of the image, which asks whether the given ordered triple is a solution of the system.

### Problem 1:
Determine if \((1, 1, 1)\) is a solution for:
\[
\begin{aligned}
x + y + z &= 3 \\
2x - y + 4z &= 5 \\
x + 4y - 2z &= 3
\end{aligned}
\]

Substitute \(x = 1\), \(y = 1\), and \(z = 1\) into each equation:

1. \(1 + 1 + 1 = 3\), which is true.
2. \(2(1) - 1 + 4(1) = 2 - 1 + 4 = 5\), which is true.
3. \(1 + 4(1) - 2(1) = 1 + 4 - 2 = 3\), which is true.

So, \((1, 1, 1)\) **is** a solution.

---

### Problem 2:
Determine if \((0, 3, 1)\) is a solution for:
\[
\begin{aligned}
x + 2y + z &= 7 \\
4x - y + 3z &= 0 \\
-2x + y - 5z &= -2
\end{aligned}
\]

Substitute \(x = 0\), \(y = 3\), and \(z = 1\) into each equation:

1. \(0 + 2(3) + 1 = 6 + 1 = 7\), which is true.
2. \(4(0) - 3 + 3(1) = -3 + 3 = 0\), which is true.
3. \(-2(0) + 3 - 5(1) = 3 - 5 = -2\), which is true.

So, \((0, 3, 1)\) **is** a solution.

---

### Problem 3:
Determine if \((2, 1, 6)\) is a solution for:
\[
\begin{aligned}
x + y - z &= -3 \\
2x - y + z &= 9 \\
4x + y - z &= 15
\end{aligned}
\]

Substitute \(x = 2\), \(y = 1\), and \(z = 6\) into each equation:

1. \(2 + 1 - 6 = 3 - 6 = -3\), which is true.
2. \(2(2) - 1 + 6 = 4 - 1 + 6 = 9\), which is true.
3. \(4(2) + 1 - 6 = 8 + 1 - 6 = 3\), but this should be 15, which is false.

So, \((2, 1, 6)\) **is not** a solution.

---

Let me know if you need details on how to solve the other problems, or if you would like to continue with the substitution method or linear combination method questions.

Here are 5 questions for you:
1. Can you explain the process of substitution used in linear systems?
2. What other methods can be used to solve linear systems apart from substitution?
3. Can the ordered triple from Problem 3 be a solution if only two equations work?
4. Why does linear combination work better for some systems?
5. What adjustments can be made if a solution does not satisfy all equations?

**Tip**: When solving systems of equations, always double-check by substituting the solutions back into **all** the equations to ensure accuracy.

Decide whether the given ordered triple is a solution of the system.

Learn how to verify if an ordered triple is a solution to a system of linear equations. Detailed explanations for ordered triples like (1, 1, 1), (0, 3, 1), and (2, 1, 6). Suitable for students in Grades 9-12.

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