Fredric leads a team of hikers for a full-day hike. The total elevation gain during the hike is 2,100 feet. All of the hikers have to pass two checkpoints before they reach the peak. The elevation gain from the starting point to checkpoint 1 is 100 feet less than double the elevation gain from checkpoint 2 to the peak. The elevation gain from checkpoint 1 to checkpoint 2 is the mean of the elevation gain from the start to checkpoint 1 and the elevation gain from checkpoint 2 to the peak.
Let [tex][tex]$x$[/tex][/tex] represent the elevation gain from the starting point to checkpoint 1, [tex][tex]$y$[/tex][/tex] represent the elevation gain from checkpoint 1 to checkpoint 2, and [tex][tex]$z$[/tex][/tex] represent the elevation gain from checkpoint 2 to the peak.
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:
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.
