solve: w-5z=-3, x+y+2z=-6, 0=0 solution is infinite but what is the written solution

Given the system of equations:

1. \( w - 5z = -3 \)
2. \( x + y + 2z = -6 \)
3. \( 0 = 0 \)

We know that the third equation \( 0 = 0 \) is always true, indicating that the system has infinite solutions, meaning there are free variables. To express the solution in terms of free variables, let's solve the system step by step.

### Step 1: Parameterize the free variable \( z \)
Since \( z \) is free (i.e., it can take any value), let:
\[ z = t \]
where \( t \) is a parameter that can take any real value.

### Step 2: Solve for \( w \) in terms of \( t \)
From the first equation:
\[
w - 5z = -3
\]
Substitute \( z = t \):
\[
w - 5t = -3 \quad \Rightarrow \quad w = 5t - 3
\]

### Step 3: Solve for \( x + y \) in terms of \( t \)
From the second equation:
\[
x + y + 2z = -6
\]
Substitute \( z = t \):
\[
x + y + 2t = -6 \quad \Rightarrow \quad x + y = -6 - 2t
\]
This equation represents a relationship between \( x \) and \( y \), meaning both are dependent on each other and \( t \).

### Final Solution:
The system has infinite solutions, which can be written as:
\[
w = 5t - 3, \quad z = t, \quad x + y = -6 - 2t
\]
where \( t \) is a free parameter. This form describes the family of solutions depending on the value of \( t \).

Would you like further details on parameterization or have any specific questions?

### Related Questions:
1. What does it mean for a system to have infinite solutions?
2. How do you identify free variables in a system of equations?
3. Can infinite solutions occur if all the equations are linearly dependent?
4. How do you interpret the relationship between \( x \) and \( y \) in this solution?
5. What would happen if we add a third independent equation to the system?

**Tip:** When dealing with infinite solutions, free variables act as parameters that define the family of all possible solutions.

solve: w - 5z = -3, x + y + 2z = -6, 0 = 0 solution is infinite but what is the written solution

Learn how to solve the system of equations w - 5z = -3 and x + y + 2z = -6 with infinite solutions. This guide covers linear algebra concepts like free variables and parameterization. Suitable for students in Grades 10-12.

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