Linear algebra 1.1 number 12 ### Chapter 1: Linear Equations in Linear Algebra#### 1.1 ExercisesSolve each system in Exercises 1–4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section.1. \[\begin{cases}x_1 + 5x_2 = 7 \\-2x_1 - 7x_2 = -5 \end{cases}\]2. \[\begin{cases}3x_1 + 6x_2 = -3 \\5x_1 + 7x_2 = 10\end{cases}\]3. Find the point $(x_1, x_2)$ that lies on the line $ x_1 + 2x_2 = 4 $ and on the line $ x_1 - x_2 = 1 $. See the figure.(Provided is a graph with two lines intersecting at point $ (2, 1) $, the lines are $ x_1 + 2x_2 = 4 $ and $ x_1 - x_2 = 1 $.)4. Find the point of intersection of the lines $ x_1 + 2x_2 = -13 $ and $ 3x_1 - 2x_2 = 5 $.Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.5. \[\begin{bmatrix}0 & 1 & -4 & 3 \ | \ 0 \\0 & 0 & 2 & 0 \ | \ 6 \\0 & 0 & 0 & 1 \ | \ -5 \end{bmatrix}\]6.\[\begin{bmatrix}1 & 0 & 6 & 4 \ | \ 1 \\0 & 1 & -7 & 0 \ | \ 0 \\0 & 0 & 1 & 4 \ | \ 2 \\0 & 0 & 0 & 2 \ | \ 3 \end{bmatrix}\]In Exercises 7–10, the

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Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Linear algebra 1.1 number 12

### Chapter 1: Linear Equations in Linear Algebra

#### 1.1 Exercises

Solve each system in Exercises 1–4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section.

1.
\[
\begin{cases}
x_1 + 5x_2 = 7 \\
-2x_1 - 7x_2 = -5
\end{cases}
\]

2.
\[
\begin{cases}
3x_1 + 6x_2 = -3 \\
5x_1 + 7x_2 = 10
\end{cases}
\]

3. Find the point $(x_1, x_2)$ that lies on the line $ x_1 + 2x_2 = 4 $ and on the line $ x_1 - x_2 = 1 $. See the figure.

(Provided is a graph with two lines intersecting at point $ (2, 1) $, the lines are $ x_1 + 2x_2 = 4 $ and $ x_1 - x_2 = 1 $.)

4. Find the point of intersection of the lines $ x_1 + 2x_2 = -13 $ and $ 3x_1 - 2x_2 = 5 $.

Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.

5.
\[
\begin{bmatrix}
0 & 1 & -4 & 3 \ | \ 0 \\
0 & 0 & 2 & 0 \ | \ 6 \\
0 & 0 & 0 & 1 \ | \ -5
\end{bmatrix}
\]

6.
\[
\begin{bmatrix}
1 & 0 & 6 & 4 \ | \ 1 \\
0 & 1 & -7 & 0 \ | \ 0 \\
0 & 0 & 1 & 4 \ | \ 2 \\
0 & 0 & 0 & 2 \ | \ 3
\end{bmatrix}
\]

In Exercises 7–10, the

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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