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Select the correct augmented matrices.

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]$x$[/tex] represent the elevation gain from the starting point to checkpoint 1, [tex]$y$[/tex] represent the elevation gain from checkpoint 1 to checkpoint 2, and [tex]$z$[/tex] represent the elevation gain from checkpoint 2 to the peak.

Which augmented matrices represent the context of this scenario?

[tex]\[
\begin{array}{ll}
\left[\begin{array}{ccc|c}
1 & 0 & -1 & 200 \\
0 & 1 & 0 & 700 \\
0 & 0 & 3 & 1,800
\end{array}\right] & \left[\begin{array}{ccc|c}
1 & 1 & 1 & 2,100 \\
-1 & 0 & 2 & 100 \\
0.5 & 1 & 0.5 & 0
\end{array}\right] \\
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 2,100 \\
-1 & 0 & 2 & 100 \\
0.5 & -1 & 0.5 & 0
\end{array}\right] & \left[\begin{array}{ccc|c}
1 & 0 & 0 & 900 \\
0 & 1 & 0 & 700 \\
0 & 0 & 1 & 500
\end{array}\right] \\
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 800 \\
0 & 1 & 0 & 700 \\
0 & 0 & 1 & 600
\end{array}\right]
\end{array}
\][/tex]



Answer :

GinnyAnswer
To solve this problem, we need to form a system of linear equations that corresponds to the given conditions about the elevation gains. Let's denote:
- [tex]\( x \)[/tex] as the elevation gain from the starting point to checkpoint 1.
- [tex]\( y \)[/tex] as the elevation gain from checkpoint 1 to checkpoint 2.
- [tex]\( z \)[/tex] as the elevation gain from checkpoint 2 to the peak.

Here are the conditions given:
1. The total elevation gain is 2,100 feet.
[tex]\[ x + y + z = 2100 \][/tex]
2. The elevation gain to checkpoint 1 is 100 feet less than double the elevation gain from checkpoint 2 to the peak.
[tex]\[ x = 2z - 100 \][/tex]
3. The elevation gain from checkpoint 1 to checkpoint 2 is the mean of the elevation gains from the start to checkpoint 1 and from checkpoint 2 to the peak.
[tex]\[ y = \frac{x + z}{2} \][/tex]

We can rewrite these equations to form a system of linear equations suitable for an augmented matrix:

1. [tex]\( x + y + z = 2100 \)[/tex]
2. Rearrange [tex]\( x = 2z - 100 \)[/tex] to form :
[tex]\[ -x + 0y + 2z = 100 \][/tex]
3. Rearrange [tex]\( y = \frac{x + z}{2} \)[/tex] to form :
[tex]\[ 0.5x + y + 0.5z = 0 \][/tex]

The augmented matrices that accurately represent these conditions are:
[tex]\[ \left[ \begin{array}{ccc|c} 1 & 1 & 1 & 2100 \\ -1 & 0 & 2 & 100 \\ 0.5 & 1 & 0.5 & 0 \end{array} \right] \][/tex]
And for the proper representation of the equations:
[tex]\[ \left[ \begin{array}{ccc|c} 1 & 1 & 1 & 2100 \\ -1 & 0 & 2 & 100 \\ 0.5 & -1 & 0.5 & 0 \end{array} \right] \][/tex]

Thus, the correct augmented matrices are:
[tex]\[ \left[ \begin{array}{ccc|c} 1 & 1 & 1 & 2100 \\ -1 & 0 & 2 & 100 \\ 0.5 & 1 & 0.5 & 0 \end{array} \right], \quad \text{and} \quad \left[ \begin{array}{ccc|c} 1 & 1 & 1 & 2100 \\ -1 & 0 & 2 & 100 \\ 0.5 & -1 & 0.5 & 0 \end{array} \right] \][/tex]

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