Let's break down the given problem and derive the system of equations that describes the elevation gains. We will then convert these equations into augmented matrices.
### Step-by-Step Solution
1. Define Variables:
- Let [tex]\( x \)[/tex] be the elevation gain from the start to checkpoint 1.
- Let [tex]\( y \)[/tex] be the elevation gain from checkpoint 1 to checkpoint 2.
- Let [tex]\( z \)[/tex] be the elevation gain from checkpoint 2 to the peak.
2. Translate the Problem into Equations:
- Total elevation gain:
[tex]\[ x + y + z = 2100 \][/tex]
- Elevation gain from start to checkpoint 1:
[tex]\[ x = 2z - 100 \][/tex]
- Elevation gain from checkpoint 1 to checkpoint 2:
[tex]\[ y = \frac{x + z}{2} \][/tex]
3. Reformulate Those Equations:
Transform the equations to a form that fits an augmented matrix:
- The first equation is:
[tex]\[ x + y + z = 2100 \][/tex]
- The second equation:
[tex]\[ x - 2z = -100 \][/tex]
- The third equation:
[tex]\[ y = \frac{x + z}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ 2y = x + z \][/tex]
Rearrange to:
[tex]\[ x - 2y + z = 0 \][/tex]
4. Form the Augmented Matrix:
Combine these equations into an augmented matrix:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 2100 \\
1 & 0 & -2 & -100 \\
1 & -2 & 1 & 0
\end{array}\right]
\][/tex]
After simplifying, we observe that the matrix format should be the following:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 2100 \\
1 & 0 & -2 & -100 \\
0.5 & -1 & 0.5 & 0
\end{array}\right]
\][/tex]
### Verify the Given Options:
Out of the given matrices, the correct augmented matrix corresponding to our derived system is:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 2100 \\
-1 & 0 & 2 & 100 \\
0.5 & -1 & 0.5 & 0
\end{array}\right]
\][/tex]
Hence, we can conclude that the correct matrices among the given options representing the context of the problem are those which match the system of equations we have derived.
