x − y + 3 z = 0
3 x + 2 y + 9 z = 15
x + 4 y + 3 z = k

You have a system of three equations with three unknowns (\(x\), \(y\), \(z\)), and the parameter \(k\). The system is:

\[
1. \ x − y + 3z = 0
\]
\[
2. \ 3x + 2y + 9z = 15
\]
\[
3. \ x + 4y + 3z = k
\]

To solve this system, we can use methods like substitution, elimination, or matrix methods (like Gaussian elimination). Let's proceed step-by-step using elimination.

### Step 1: Express \(x\) from Equation (1)
From equation (1), solve for \(x\):
\[
x = y - 3z
\]

### Step 2: Substitute \(x = y - 3z\) into Equations (2) and (3)
#### Substitute into Equation (2):
\[
3(y - 3z) + 2y + 9z = 15
\]
Simplify:
\[
3y - 9z + 2y + 9z = 15
\]
\[
5y = 15
\]
\[
y = 3
\]

#### Substitute into Equation (3):
\[
(y - 3z) + 4y + 3z = k
\]
Substitute \(y = 3\):
\[
(3 - 3z) + 12 + 3z = k
\]
Simplify:
\[
3 + 12 = k
\]
\[
k = 15
\]

### Step 3: Solve for \(z\)
Now that we know \(y = 3\), substitute this back into Equation (1):
\[
x - 3 + 3z = 0
\]
\[
x = 3 - 3z
\]
Substitute \(x = 3 - 3z\) and \(y = 3\) into Equation (2):
\[
3(3 - 3z) + 2(3) + 9z = 15
\]
Simplify:
\[
9 - 9z + 6 + 9z = 15
\]
\[
15 = 15
\]

This is true, meaning the system is consistent for any \(z\).

### Final Solution
The solution depends on \(z\), so the system has infinitely many solutions. The solutions are given by:
\[
x = 3 - 3z, \quad y = 3, \quad z = z, \quad k = 15
\]
The parameter \(z\) can take any real value, and \(x\) depends on \(z\).

Would you like more details on a specific step or method?

Here are 5 related questions:
1. What is Gaussian elimination and how does it apply to solving systems of equations?
2. How do you solve systems of equations using matrix methods (like row reduction)?
3. What is the significance of having a parameter in the solution of a system of equations?
4. How do you classify a system of equations (consistent, inconsistent, dependent)?
5. What are alternative methods for solving systems of linear equations?

**Tip:** Always check the determinant when dealing with square systems (3 equations, 3 unknowns) to determine if the system has unique solutions.

x − y + 3z = 0, 3x + 2y + 9z = 15, x + 4y + 3z = k

Learn how to solve a system of three linear equations involving parameters. This step-by-step solution covers key algebraic concepts, including substitution and consistency in linear systems. Explore how parameter-dependent solutions arise, with the solution set expressed in terms of z. Suitable for students in Grades 9-12.

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