Difference between revisions of "031 Review Part 2, Problem 11"

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<span class="exam">Find all real values of &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; such that the system has only one solution.
 
<span class="exam">Find all real values of &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; such that the system has only one solution.
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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           1 & k & 1 \\
 
           1 & k & 1 \\
 
           3 & 5  & 2k   
 
           3 & 5  & 2k   
 +
        \end{array}\right].</math>
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|-
 +
|Now, when we row reduce this matrix, we get
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|-
 +
|
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::<math>\left[\begin{array}{cc|c} 
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          1 & k & 1 \\
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          3 & 5  & 2k 
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        \end{array}\right] \sim \left[\begin{array}{cc|c} 
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          1 & k & 1 \\
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          0 & -3k+5  & -3+2k 
 
         \end{array}\right].</math>
 
         \end{array}\right].</math>
 
|}
 
|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
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|-
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|To guarantee a unique solution, our matrix must contain two pivots.
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|-
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|So, we must have &nbsp;<math style="vertical-align: -5px">-3k+5\ne 0.</math>
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|-
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|Hence, we must have
 
|-
 
|-
 
|
 
|
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::<math>k\ne \frac {5}{3}.</math>
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|-
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|Therefore, &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; can be any real number except &nbsp;<math style="vertical-align: -13px">\frac{5}{3}.</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|-
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|&nbsp;&nbsp; &nbsp; &nbsp;  
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|&nbsp;&nbsp; &nbsp; &nbsp; The system has only one solution when &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; is any real number except &nbsp;<math style="vertical-align: -13px">\frac{5}{3}.</math>
 
|}
 
|}
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[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
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[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

Latest revision as of 13:43, 15 October 2017

Consider the following system of equations.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3x_1+5x_2=2k}

Find all real values of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}   such that the system has only one solution.

Foundations:  
1. To solve a system of equations, we turn the system into an augmented matrix and
row reduce that matrix to determine the solution.
2. For a system to have a unique solution, we need to have no free variables.


Solution:

Step 1:  
To begin with, we turn this system into an augmented matrix.
Hence, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\begin{array}{cc|c} 1 & k & 1 \\ 3 & 5 & 2k \end{array}\right].}
Now, when we row reduce this matrix, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\begin{array}{cc|c} 1 & k & 1 \\ 3 & 5 & 2k \end{array}\right] \sim \left[\begin{array}{cc|c} 1 & k & 1 \\ 0 & -3k+5 & -3+2k \end{array}\right].}
Step 2:  
To guarantee a unique solution, our matrix must contain two pivots.
So, we must have  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3k+5\ne 0.}
Hence, we must have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\ne \frac {5}{3}.}
Therefore,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}   can be any real number except  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5}{3}.}


Final Answer:  
       The system has only one solution when  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}   is any real number except  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5}{3}.}

Return to Review Problems

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