The matrix 

1	−1	2	

0	0	0
 is in reduced row-echelon form. Determine the solution of the corresponding system of linear equations. Write the solution as an ordered pair. For parametric solutions use either 
x = t
 or 
y = t
 as the parameter. (If an answer does not exist, enter DNE.)
(x, y) =

The matrix given is:

\[
\begin{bmatrix}
1 & -1 & 2 \\
0 & 0 & 0
\end{bmatrix}
\]

This represents a system of linear equations. Each row corresponds to an equation in terms of \(x\) and \(y\).

### Step 1: Interpret the first row
The first row is:
\[
1x - 1y = 2
\]
or
\[
x - y = 2
\]

### Step 2: Solve for \(x\)
Rearranging the equation:
\[
x = y + 2
\]

### Step 3: Parametric form
Since there is no second equation (the second row is all zeros), \(y\) can take any value. Let \(y = t\), where \(t\) is a parameter. Then:
\[
x = t + 2
\]

### Solution as an ordered pair:
\[
(x, y) = (t + 2, t)
\]

Thus, the solution to the system is \( (x, y) = (t + 2, t) \), where \(t\) is a free parameter.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How do we determine if a matrix is in reduced row-echelon form?
2. What does it mean for a system of equations to have infinitely many solutions?
3. Can you generalize this method for larger systems of equations?
4. How can you graph the parametric solution \( (x, y) = (t + 2, t) \)?
5. What are the implications of having a row of zeros in a matrix?

**Tip:** In systems with free variables, introducing a parameter like \( t \) helps to describe the infinitely many solutions systematically.

The matrix 
1 -1 2 
0 0 0 
is in reduced row-echelon form. Determine the solution of the corresponding system of linear equations. Write the solution as an ordered pair. For parametric solutions use either x = t or y = t as the parameter. (If an answer does not exist, enter DNE.) (x, y) =

Understand how to solve a system of linear equations using a reduced row-echelon matrix. This problem involves key concepts in linear algebra, including parametric solutions and free variables. Perfect for high school students in Grades 9-12.

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