005 Sample Final A, Question 9 - Revision history

Matt Lee at 03:29, 22 May 2015

2015-05-22T03:29:11Z

Matt Lee at 06:48, 19 May 2015

2015-05-19T06:48:53Z

Matt Lee at 06:47, 19 May 2015

2015-05-19T06:47:38Z

Matt Lee at 06:47, 19 May 2015

2015-05-19T06:47:25Z

Matt Lee: Created page with "''' Question ''' Solve the following system of equations
$\begin{align} 2x + 3y &= & 1\\ -x + y & = & -3\end{align}$ {| class="mw-collaps..."

2015-05-01T04:23:35Z

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..."

New page

''' Question ''' Solve the following system of equations <br>
<center><math> \begin{align} 2x + 3y &= & 1\\ -x + y & = & -3\end{align}</math></center>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answers
|-
|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>
|-
|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.
|-
|c) False. <math>y = x^2</math> does not have an inverse.
|-
|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
|-
|e) True.
|-
|f) False.
|}

← Older revision		Revision as of 03:29, 22 May 2015
Line 1:		Line 1:
	''' Question ''' Solve the following system of equations <br>		''' Question ''' Solve the following system of equations <br>
	::<math> \begin{align} 2x + 3y &= & 1\\ -x + y & = & -3\end{align}</math>		::<math> \begin{align} 2x + 3y &= & 1\\ -x + y & = & -3\end{align}</math>
		+
		+	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"
		+	!Foundations:
		+	\|-
		+	\|1) What are the two methods for solving a system of equations?
		+	\|-
		+	\|2) How do we use the substitution method?
		+	\|-
		+	\|3) How do we use the elimination method?
		+	\|-
		+	\|Answer:
		+	\|-
		+	\|1) The two methods are the substitution and elimination methods.
		+	\|-
		+	\|2) Solve for x or y in one of the equations and substitute that value into the other equation.
		+	\|-
		+	\|3) Multiply one equation by some number on both sides to make one of the variables, x or y, have the same coefficient and add the equations together.
		+	\|}

@@ Line 26: / Line 26: @@
+|
 <math>\begin{array}{rcl}
--x -1 &=& -4\\
+-x -1 &=& -3\\
--x & =& -3\\
+-x & =& -2\\
-x&=&3
+x&=&2
 \end{array}</math>
 |}
@@ Line 35: / Line 35: @@
 ! Final Answer:
 |-
-|<math>x = 3,~ y = -1</math>
+|<math>x = 2,~ y = -1</math>
 |}

@@ Line 35: / Line 35: @@
 ! Final Answer:
 |-
-|<math>x = 3, y = -1</math>
+|<math>x = 3,~ y = -1</math>
 |}

@@ Line 1: / Line 1: @@
 ''' Question ''' Solve the following system of equations <br>
-<center><math>  \begin{align} 2x + 3y  &= & 1\\ -x + y & = & -3\end{align}</math></center>
+::<math>  \begin{align} 2x + 3y  &= & 1\\ -x + y & = & -3\end{align}</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Final Answers
+! Step 1:
 |-
-|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>
+|Add two times the second equation to the first equation. So we are adding <math>-2x + 2y = -6</math> to the first equation.
 |-
-|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.
+|This leads to:
 |-
-|c) False. <math>y = x^2</math> does not have an inverse.
 |-
-|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
+|This gives us that <math>y = -1.</math>
 |-
-|e) True.
+|Now we just need to find x. So we plug in -1 for y in the second equation.
 |-
-|f) False.
+|<math>x = 3, y = -1</math>
 |}

005 Sample Final A, Question 9 - Revision history

Matt Lee at 03:29, 22 May 2015

Matt Lee at 06:48, 19 May 2015

Matt Lee at 06:47, 19 May 2015

Matt Lee at 06:47, 19 May 2015

Matt Lee: Created page with "''' Question ''' Solve the following system of equations \begin{align} 2x + 3y &= & 1\\ -x + y & = & -3\end{align} {| class="mw-collaps..."

Matt Lee: Created page with "''' Question ''' Solve the following system of equations
$\begin{align} 2x + 3y &= & 1\\ -x + y & = & -3\end{align}$ {| class="mw-collaps..."