## 数学代写|线性代数代写linear algebra代考|MAST10022

2023年1月5日

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## 数学代写|线性代数代写linear algebra代考|Balancing Chemical Equations

Chemical equations describe the quantities of substances consumed and produced by chemical reactions. For instance, when propane gas burns, the propane $\left(\mathrm{C}_3 \mathrm{H}_8\right)$ combines with oxygen $\left(\mathrm{O}_2\right)$ to form carbon dioxide $\left(\mathrm{CO}_2\right)$ and water $\left(\mathrm{H}_2 \mathrm{O}\right)$, according to an equation of the form
$$\left(x_1\right) \mathrm{C}_3 \mathrm{H}_8+\left(x_2\right) \mathrm{O}_2 \rightarrow\left(x_3\right) \mathrm{CO}_2+\left(x_4\right) \mathrm{H}_2 \mathrm{O}$$
To “balance” this equation, a chemist must find whole numbers $x_1, \ldots, x_4$ such that the total numbers of carbon $(\mathrm{C})$, hydrogen $(\mathrm{H})$, and oxygen $(\mathrm{O})$ atoms on the left match the corresponding numbers of atoms on the right (because atoms are neither destroyed nor created in the reaction).

A systematic method for balancing chemical equations is to set up a vector equation that describes the numbers of atoms of each type present in a reaction. Since equation (4) involves three types of atoms (carbon, hydrogen, and oxygen), construct a vector in $\mathbb{R}^3$ for each reactant and product in (4) that lists the numbers of “atoms per molecule,” as follows:
To balance equation (4), the coefficients $x_1, \ldots, x_4$ must satisfy
$$x_1\left[\begin{array}{l} 3 \ 8 \ 0 \end{array}\right]+x_2\left[\begin{array}{l} 0 \ 0 \ 2 \end{array}\right]=x_3\left[\begin{array}{l} 1 \ 0 \ 2 \end{array}\right]+x_4\left[\begin{array}{l} 0 \ 2 \ 1 \end{array}\right]$$
To solve, move all the terms to the left (changing the signs in the third and fourth vectors):
$$x_1\left[\begin{array}{l} 3 \ 8 \ 0 \end{array}\right]+x_2\left[\begin{array}{l} 0 \ 0 \ 2 \end{array}\right]+x_3\left[\begin{array}{r} -1 \ 0 \ -2 \end{array}\right]+x_4\left[\begin{array}{r} 0 \ -2 \ -1 \end{array}\right]=\left[\begin{array}{l} 0 \ 0 \ 0 \end{array}\right]$$
Row reduction of the augmented matrix for this equation leads to the general solution
$$x_1=\frac{1}{4} x_4, x_2=\frac{5}{4} x_4, x_3=\frac{3}{4} x_4 \text {, with } x_4 \text { free }$$
Since the cocfficients in a chemical cquation must be integers, take $x_4=4$, in which case $x_1=1, x_2=5$, and $x_3=3$. The halanced equation is
$$\mathrm{C}_3 \mathrm{H}_8+5 \mathrm{O}_2 \rightarrow 3 \mathrm{CO}_2+4 \mathrm{H}_2 \mathrm{O}$$
The equation would also be balanced if, for example, each coefficient were doubled. For most purposes, however, chemists prefer to use a balanced equation whose coefficients are the smallest possible whole numbers.

## 数学代写|线性代数代写linear algebra代考|Network Flow

Systems of linear equations arise naturally when scientists, engineers, or economists study the flow of some quantity through a network. For instance, urban planners and traffic engineers monitor the pattern of traffic flow in a grid of city streets. Electrical engineers calculate current flow through electrical circuits. And economists analyze the distribution of products from manufacturers to consumers through a network of wholesalers and retailers. For many networks, the systems of equations involve hundreds or even thousands of variables and equations.

A network consists of a set of points called junctions, or nodes, with lines or arcs called branches connecting some or all of the junctions. The direction of flow in each branch is indicated, and the flow amount (or rate) is either shown or is denoted by a variable.

The basic assumption of network flow is that the total flow into the network equals the total flow out of the network and that the total flow into a junction equals the total flow out of the junction. For example, Figure 1 shows 30 units flowing into a junction through one branch, with $x_1$ and $x_2$ denoting the flows out of the junction through other branches. Since the flow is “conserved” at each junction, we must have $x_1+x_2=30$. In a similar fashion, the flow at each junction is described by a linear equation. The problem of network analysis is to determine the flow in each branch when partial information (such as the flow into and out of the network) is known.

EXAMPLE 2 The network in Figure 2 shows the traffic flow (in vehicles per hour) over several one-way streets in downtown Baltimore during a typical early afternoon. Determine the general flow pattern for the network.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Balancing Chemical Equations

$x_1, \ldots, x_4$ 这样碳的总数 $(\mathrm{C})$, 氢气 $(\mathrm{H})$, 和氧气 $(\mathrm{O})$ 左 边的原子与右边原子的相应数量相匹配 (因为原子在反 应中既没有被破坏也没有被创造）。

$$x_1\left[\begin{array}{lll} 3 & 8 & 0 \end{array}\right]+x_2\left[\begin{array}{lll} 0 & 0 & 2 \end{array}\right]=x_3\left[\begin{array}{lll} 1 & 0 & 2 \end{array}\right]+x_4\left[\begin{array}{lll} 0 & 2 & 1 \end{array}\right]$$

$$x_1\left[\begin{array}{lll} 3 & 8 & 0 \end{array}\right]+x_2\left[\begin{array}{lll} 0 & 0 & 2 \end{array}\right]+x_3\left[\begin{array}{lll} -1 & 0 & -2 \end{array}\right]+x_4[0$$

$x_1=\frac{1}{4} x_4, x_2=\frac{5}{4} x_4, x_3=\frac{3}{4} x_4$, with $x_4$ free

$$\mathrm{C}_3 \mathrm{H}_8+5 \mathrm{O}_2 \rightarrow 3 \mathrm{CO}_2+4 \mathrm{H}_2 \mathrm{O}$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。