Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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.
### 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.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.