Difference between revisions of "005 Sample Final A, Question 9"

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(Created page with "''' Question ''' Solve the following system of equations <br> <center><math> \begin{align} 2x + 3y &= & 1\\ -x + y & = & -3\end{align}</math></center> {| class="mw-collaps...")
 
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''' Question ''' Solve the following system of equations <br>
 
''' Question ''' Solve the following system of equations <br>
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<center><math>  \begin{align} 2x + 3y  &= & 1\\ -x + y & = & -3\end{align}</math></center>
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::<math>  \begin{align} 2x + 3y  &= & 1\\ -x + y & = & -3\end{align}</math>
  
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answers
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! Step 1:
 
|-
 
|-
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|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
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|Add two times the second equation to the first equation. So we are adding <math>-2x + 2y = -6</math> to the first equation.
 
|-
 
|-
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|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
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|This leads to:
 
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|-
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|c) False. <math>y = x^2</math> does not have an inverse.
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|
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::<math>\begin{array}{rcl}
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0 + 5y &=& -5\\
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-x + y &=& -3
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\end{array}</math>
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 2:
 
|-
 
|-
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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|This gives us that <math>y = -1.</math>
 
|-
 
|-
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|e) True.
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|Now we just need to find x. So we plug in -1 for y in the second equation.
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|-
 +
|
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<math>\begin{array}{rcl}
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-x -1 &=& -4\\
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-x & =& -3\\
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x&=&3
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\end{array}</math>
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|}
 +
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answer:
 
|-
 
|-
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|f) False.
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|<math>x = 3, y = -1</math>
 
|}
 
|}

Revision as of 22:47, 18 May 2015

Question Solve 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 \begin{align} 2x + 3y &= & 1\\ -x + y & = & -3\end{align}}


Step 1:
Add two times the second equation to the first equation. So we are adding 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 -2x + 2y = -6} to the first equation.
This leads to:
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 \begin{array}{rcl} 0 + 5y &=& -5\\ -x + y &=& -3 \end{array}}
Step 2:
This gives us that 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 y = -1.}
Now we just need to find x. So we plug in -1 for y in the second equation.

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 \begin{array}{rcl} -x -1 &=& -4\\ -x & =& -3\\ x&=&3 \end{array}}

Final Answer:
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 x = 3, y = -1}
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