homework help with linear algebra, thank you! i can't get the right answer!
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Transcribed Image Text:### Solving Linear Systems
In this task, we are asked to find all the solutions to a given linear system. The system is represented in augmented matrix form as follows:
\[
\begin{bmatrix}
1 & 1 & -1 \\
0 & 1 & -1 \\
\end{bmatrix}
\begin{bmatrix}
c_1 \\
c_2
\end{bmatrix}
=
\begin{bmatrix}
4 \\
0
\end{bmatrix}
\]
This system can be broken down into the following linear equations:
1. \(1 \cdot c_1 + 1 \cdot c_2 = 4\)
2. \(0 \cdot c_1 + 1 \cdot c_2 = 0\)
The task is to solve these equations for the variables \(c_1\) and \(c_2\).
### Steps to Solve
1. **From the Second Equation**: Since \(0 \cdot c_1 + 1 \cdot c_2 = 0\), we have \(c_2 = 0\).
2. **Substitute \(c_2\) into the First Equation**: Substitute \(c_2 = 0\) into the first equation, \(1 \cdot c_1 + 1 \cdot 0 = 4\), which simplifies to \(c_1 = 4\).
Therefore, the solution to the system is:
- \(c_1 = 4\)
- \(c_2 = 0\)
This provides the unique solution for the given linear equations.
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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![### Solving Linear Systems
In this task, we are asked to find all the solutions to a given linear system. The system is represented in augmented matrix form as follows:
\[
\begin{bmatrix}
1 & 1 & -1 \\
0 & 1 & -1 \\
\end{bmatrix}
\begin{bmatrix}
c_1 \\
c_2
\end{bmatrix}
=
\begin{bmatrix}
4 \\
0
\end{bmatrix}
\]
This system can be broken down into the following linear equations:
1. \(1 \cdot c_1 + 1 \cdot c_2 = 4\)
2. \(0 \cdot c_1 + 1 \cdot c_2 = 0\)
The task is to solve these equations for the variables \(c_1\) and \(c_2\).
### Steps to Solve
1. **From the Second Equation**: Since \(0 \cdot c_1 + 1 \cdot c_2 = 0\), we have \(c_2 = 0\).
2. **Substitute \(c_2\) into the First Equation**: Substitute \(c_2 = 0\) into the first equation, \(1 \cdot c_1 + 1 \cdot 0 = 4\), which simplifies to \(c_1 = 4\).
Therefore, the solution to the system is:
- \(c_1 = 4\)
- \(c_2 = 0\)
This provides the unique solution for the given linear equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47978d15-e20a-4071-98c6-506c77d70e17%2Fab2ba843-0a16-4f9a-a31a-71b98f216f87%2Fumbtjow.jpeg&w=3840&q=75)
