4. Solve the following linear systems. 21 + 2x2 2x1 + x1 (a). - 22 x2 x3 3 + 24 = 4 x4 1 + x3 +

Calculus: Early Transcendentals
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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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## Solving Linear Systems

### Problem 4: Solve the following linear systems.

#### (a)
\[
\begin{cases} 
x_1 + 2x_2 - x_3 = 3 \\
2x_1 + x_2 + x_4 = 4 \\
x_1 - x_2 + x_3 + x_4 = 1 
\end{cases}
\]

#### (b)
\[
\begin{cases} 
x + 2x_2 + 3x_3 + x_4 - x_5 = 1 \\
3x_1 - x_2 + x_3 + x_4 + x_5 = 3 
\end{cases}
\]

### Explanation

**Part (a)** consists of a system of three linear equations with four unknowns: \(x_1, x_2, x_3, x_4\). Each equation represents a linear combination of these variables set equal to a constant.

**Part (b)** features two linear equations with five unknowns: \(x, x_2, x_3, x_4, x_5\). These equations also represent linear combinations of the unknowns, each equated to a constant. 

To solve these systems, techniques such as substitution, elimination, or matrix methods (like Gaussian elimination) can be utilized to find the values of the unknown variables that satisfy all equations simultaneously.
Transcribed Image Text:## Solving Linear Systems ### Problem 4: Solve the following linear systems. #### (a) \[ \begin{cases} x_1 + 2x_2 - x_3 = 3 \\ 2x_1 + x_2 + x_4 = 4 \\ x_1 - x_2 + x_3 + x_4 = 1 \end{cases} \] #### (b) \[ \begin{cases} x + 2x_2 + 3x_3 + x_4 - x_5 = 1 \\ 3x_1 - x_2 + x_3 + x_4 + x_5 = 3 \end{cases} \] ### Explanation **Part (a)** consists of a system of three linear equations with four unknowns: \(x_1, x_2, x_3, x_4\). Each equation represents a linear combination of these variables set equal to a constant. **Part (b)** features two linear equations with five unknowns: \(x, x_2, x_3, x_4, x_5\). These equations also represent linear combinations of the unknowns, each equated to a constant. To solve these systems, techniques such as substitution, elimination, or matrix methods (like Gaussian elimination) can be utilized to find the values of the unknown variables that satisfy all equations simultaneously.
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