Difference between revisions of "Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 1 (a)/Solution 1"

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Note that <math>x_1, x_2</math> and <math>x_3</math> denote the amount of money Uno, Duo, and Traea owe, respectively.
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Note that <math>x_1</math>, <math>x_2</math>, and <math>x_3</math> denote the amount of money Uno, Duo, and Traea owe, respectively.
  
We first express three information "(1) All together they owe $600", "(2) Duo owes $200 more than Uno", (3)"Uno and Duo combined owe as much as Traea" to equations of <math>x_1</math>, <math>x_2</math>, and <math>x_3</math>;
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We first express the three statements (1) "All together they owe $600", (2) "Duo owes $200 more than Uno", and (3) "Uno and Duo combined owe as much as Traea" as linear equations in <math>x_1</math>, <math>x_2</math>, and <math>x_3</math>:
  
<math display="block">\begin{cases}x_1+x_2+x_3=600 \  \text{by}\  (1)\\x_2-x_1=200 \ \text{by}\ (2)\\ x_1+x_2=x_3\ \text{by}\ (3)\end{cases}</math>  
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<math display="block">\begin{cases}x_1+x_2+x_3=600 & (1)\\x_2-x_1=200 & (2)\\ x_1+x_2=x_3 & (3)\end{cases}.</math>  
  
 
This system can be rewritten as
 
This system can be rewritten as
  
<math display="block">\begin{align}\begin{cases}x_1+x_2+x_3=600 \\-x_1+x_2+0\cdot x_3=200 \\ x_1+x_2-x_3=0\end{cases}\end{align}</math>  
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<math display="block">\begin{cases}x_1 +x_2+x_3=600 \\-x_1 +x_2+0x_3=200 \\ x_1 +x_2-x_3=0\end{cases}.</math>  
  
  
Then, the coefficient matrix <math>A</math> can be formed by aligning the coefficients of the variables of each equation in a row. For the constant matrix <math>b</math>, we keep the order;
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Then, the coefficient matrix <math>A</math> can be formed by reading off the coefficients of each variable in each row. For the constant vector <math>\mathbf{b}</math>, we keep the same values:
  
<math display="block">\begin{pmatrix}1&1&1\\-1&1&0\\1&1&-1 \end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3 \end{pmatrix}=\begin{pmatrix}600\\200\\0 \end{pmatrix}</math>  
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<math display="block">\begin{bmatrix}1&1&1\\-1&1&0\\1&1&-1 \end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix}=\begin{bmatrix}600\\200\\0 \end{bmatrix}.</math>  
  
Therefore, we find
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Therefore, we find that
  
<math display="block">A = \begin{pmatrix}1&1&1\\-1&1&0\\1&1&-1 \end{pmatrix}, \ \ b=\begin{pmatrix}600\\200\\0 \end{pmatrix}</math>  
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<math display="block">\color{blue} A = \begin{bmatrix}1&1&1\\-1&1&0\\1&1&-1 \end{bmatrix},\quad \mathbf{b}=\begin{bmatrix}600\\200\\0 \end{bmatrix}.</math>  
  
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Note that the rows of the matrix <math>A</math> and vector <math>\mathbf{b}</math> may be freely permuted without affecting the system; thus, for instance,
  
Changing the order of first two equations doesn't affect on the system;
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<math display="block">A = \begin{bmatrix}-1&1&0\\1&1&1\\1&1&-1 \end{bmatrix},\quad \mathbf{b}=\begin{bmatrix}200\\600\\0 \end{bmatrix}.</math>
  
<math display="block">\begin{align}\begin{cases}-x_1+x_2+0\cdot x_3=200 \\x_1+x_2+x_3=600 \\ x_1+x_2-x_3=0\end{cases}\end{align}</math>
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is also a valid answer.
 
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But their matrix form is changed;
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<math display="block">\begin{pmatrix}-1&1&0\\1&1&1\\1&1&-1 \end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3 \end{pmatrix}=\begin{pmatrix}200\\600\\0 \end{pmatrix}</math>
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Thus we find another answer for <math>(A,b)</math> pair and all similar variations could be the answer;
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<math display="block">\color{blue}A=\begin{pmatrix}-1&1&0\\1&1&1\\1&1&-1 \end{pmatrix} \ \ b=\begin{pmatrix}200\\600\\0 \end{pmatrix}.</math>
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Latest revision as of 20:04, 8 March 2018

Note that , , and denote the amount of money Uno, Duo, and Traea owe, respectively.

We first express the three statements (1) "All together they owe $600", (2) "Duo owes $200 more than Uno", and (3) "Uno and Duo combined owe as much as Traea" as linear equations in , , and :

This system can be rewritten as


Then, the coefficient matrix can be formed by reading off the coefficients of each variable in each row. For the constant vector , we keep the same values:

Therefore, we find that

Note that the rows of the matrix and vector may be freely permuted without affecting the system; thus, for instance,

is also a valid answer.