Consider this Reduced-Row Echelon Form (RREF) of the augmented matrix of a system of linear
equations:

In the entry boxes provided below, enter the solutions for variables and in terms of the
variables and (one digit per box), where you will need to determine the coefficients and
the constants in each equation.

Ask by Fleming Watson. in the United States

Jan 27,2025

Question

Consider this Reduced-Row Echelon Form (RREF) of the augmented matrix of a system of linear
equations:

In the entry boxes provided below, enter the solutions for variables and in terms of the
variables and (one digit per box), where you will need to determine the coefficients and
the constants in each equation.

Ask by Fleming Watson. in the United States

Jan 27,2025

UpStudy AI · Accepted Answer

Given the Reduced-Row Echelon Form (RREF) of the augmented matrix:

$$
\left[\begin{array}{cccc:c}
1 & 0 & 0 & 2 & 0 \
0 & 1 & 0 & 1 & 0 \
0 & 0 & 1 & -1 & 0
\end{array}\right]
$$

This corresponds to the following system of equations:

$$
\begin{cases}
x_1 + 2x_4 = 0 \
x_2 + x_4 = 0 \
x_3 - x_4 = 0
\end{cases}
$$

To express $$ x_1 $$ and $$ x_3 $$ in terms of $$ x_2 $$ and $$ x_4 $$:

1. **From the second equation:**
   $$
   x_2 + x_4 = 0 \implies x_2 = -x_4
   $$
   
2. **Substitute $$ x_2 = -x_4 $$ into the first and third equations:**
   $$
   x_1 + 2x_4 = 0 \implies x_1 = -2x_4 = 2x_2
   $$
   $$
   x_3 - x_4 = 0 \implies x_3 = x_4 = -x_2
   $$

Thus, the solutions are:

$$
\begin{align*}
x_1 &= 2x_2 + 0x_4 \
x_3 &= -1x_2 + 0x_4
\end{align*}
$$

**Final Answer:**

- $$ x_{1} = \boxed{2\,x_{2} \,+\, 0\,x_{4}} $$
- $$ x_{3} = \boxed{-1\,x_{2} \,+\, 0\,x_{4}} $$
$$ x_{1} = 2x_{2} $$ and $$ x_{3} = -x_{2} $$.

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