Systems of Equations in Two Variables - Revision history

Lila at 18:19, 21 October 2021

2021-10-21T18:19:00Z

Lila: /* Linear Simultaneous Equations with Two Variables */

2021-10-07T22:04:17Z

Linear Simultaneous Equations with Two Variables

Lila at 22:03, 7 October 2021

2021-10-07T22:03:46Z

Lila at 16:07, 15 September 2021

2021-09-15T16:07:24Z

Lila: /* Introduction */

2021-09-15T16:03:45Z

Introduction

Lila at 16:03, 15 September 2021

2021-09-15T16:03:17Z

Lila: /* Introduction */

2021-09-15T16:02:40Z

Introduction

Lila at 15:59, 15 September 2021

2021-09-15T15:59:32Z

Lila at 15:53, 15 September 2021

2021-09-15T15:53:21Z

Lila at 15:52, 15 September 2021

2021-09-15T15:52:39Z

← Older revision		Revision as of 18:19, 21 October 2021
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	* [https://www.youtube.com/watch?v=cwHR_B9zK7k Solving Systems of Equations with Substitution], patrickJMT		* [https://www.youtube.com/watch?v=cwHR_B9zK7k Solving Systems of Equations with Substitution], patrickJMT
	* [https://www.youtube.com/watch?v=WNkPKv0OTGI Solving Systems of Equations with Graphing], patrickJMT		* [https://www.youtube.com/watch?v=WNkPKv0OTGI Solving Systems of Equations with Graphing], patrickJMT
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://openstax.org/books/college-algebra/pages/7-1-systems-of-linear-equations-two-variables Systems of Linear Equations: Two Variables, OpenStax: College Algebra] under a CC BY license
		+	* [https://en.wikibooks.org/wiki/Algebra/Systems_of_Equations Systems of Equations, Wikibooks: Algebra] under a CC BY-SA license

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 * Infinite solutions: <math> y = sin(x) </math> and <math> y = 0 </math>. <math> sin(x) = 0 </math> for all <math> x = \pi k </math>, where k is some integer. So, this system has an infinite number of solutions of the form <math> (\pi k, 0) </math>; that is, the solution set for this system of equations is {.....<math> (-2\pi, 0) </math>, <math> (-\pi, 0) </math>, <math> (0, 0) </math>, <math> (\pi, 0) </math>, <math> (2\pi, 0) </math>.....}.
-==Linear Simultaneous Equations with Two Variables==
+==Linear Equations with Two Variables==
 In the previous module, linear equations with two variables were discussed.  A single linear equation having two unknown variables is practically insufficient to solve or even narrow down the solutions for the two variables, although it does establish a relationship between them.  The relationship is shown graphically as a line.  Another linear equation with the same two variables may be enough to narrow down the solution to the two equations to one value for the first variable and one value for the second variable, i. e. to solve the system of two simultaneous linear equations.  Let's see how two linear equations with the same two unknowns might be related to each other.  Since we said it was given that both equations were linear, the graphs of both equations would be lines in the same two-dimensional coordinate plane (for a system with two variables).  The lines could be related to each other in the following three ways:

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 * No solutions: <math> y = x^2 </math> and <math> y = -3 </math>. Since <math> x^2 </math> is positive for all values of x, it cannot equal -3. When graphed, we can see that these two equations never intersect. Thus there are no solutions to this system of equations.
 * Infinite solutions: <math> y = sin(x) </math> and <math> y = 0 </math>. <math> sin(x) = 0 </math> for all <math> x = \pi k </math>, where k is some integer. So, this system has an infinite number of solutions of the form <math> (\pi k, 0) </math>; that is, the solution set for this system of equations is {.....<math> (-2\pi, 0) </math>, <math> (-\pi, 0) </math>, <math> (0, 0) </math>, <math> (\pi, 0) </math>, <math> (2\pi, 0) </math>.....}.
 ==Resources==

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 * Multiple solutions: <math> y = x^2 </math> and <math> y = 4 </math>. Since <math> x^2 = 4 </math> when x = 2 or -2, we know there are two solutions to this system: (2, 4) and (-2, 4). This is also clear when we graph these two equations, as <math> y = 4 </math> intersects the parabola <math> y = x^2 </math> when x = -2, and when x = 2.
 * No solutions: <math> y = x^2 </math> and <math> y = -3 </math>. Since <math> x^2 </math> is positive for all values of x, it cannot equal -3. When graphed, we can see that these two equations never intersect. Thus there are no solutions to this system of equations.
-* Infinite solutions: <math> y = sin(x) </math> and <math> y = 0 </math>. <math> sin(x) = 0 </math> for all <math> x = \pi k </math>, where k is some integer. So, this system has an infinite number of solutions of the form <math> (\pi k, 0) </math>; that is, the solution set for this system of equations is {...<math> (-2\pi, 0) </math>, <math> (-\pi, 0) </math>, <math> (0, 0) </math>, <math> (\pi, 0) </math>, <math> (2\pi, 0) </math>...}.
+* Infinite solutions: <math> y = sin(x) </math> and <math> y = 0 </math>. <math> sin(x) = 0 </math> for all <math> x = \pi k </math>, where k is some integer. So, this system has an infinite number of solutions of the form <math> (\pi k, 0) </math>; that is, the solution set for this system of equations is {.....<math> (-2\pi, 0) </math>, <math> (-\pi, 0) </math>, <math> (0, 0) </math>, <math> (\pi, 0) </math>, <math> (2\pi, 0) </math>.....}.
 ==Resources==
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 * [https://openstax.org/books/college-algebra/pages/7-1-systems-of-linear-equations-two-variables Systems of Linear Equations with Two Variables], OpenStax
 * [https://openstax.org/books/college-algebra/pages/7-3-systems-of-nonlinear-equations-and-inequalities-two-variables Systems of Nonlinear Equations with Two Variables], OpenStax

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 * Multiple solutions: <math> y = x^2 </math> and <math> y = 4 </math>. Since <math> x^2 = 4 </math> when x = 2 or -2, we know there are two solutions to this system: (2, 4) and (-2, 4). This is also clear when we graph these two equations, as <math> y = 4 </math> intersects the parabola <math> y = x^2 </math> when x = -2, and when x = 2.
 * No solutions: <math> y = x^2 </math> and <math> y = -3 </math>. Since <math> x^2 </math> is positive for all values of x, it cannot equal -3. When graphed, we can see that these two equations never intersect. Thus there are no solutions to this system of equations.
-* Infinite solutions: <math> y = sin(x) </math> and <math> y = 0 </math>. <math> sin(x) = 0 </math> for all <math> x = \pi k </math>, where k is some integer. So, this system has an infinite number of solutions of the form <math> (\pi k, 0) </math>; that is, the solution set is {...<math> (-2\pi, 0) </math>, <math> (-\pi, 0) </math>, <math> (0, 0) </math>, <math> (\pi, 0) </math>, <math> (2\pi, 0) </math>...}.
+* Infinite solutions: <math> y = sin(x) </math> and <math> y = 0 </math>. <math> sin(x) = 0 </math> for all <math> x = \pi k </math>, where k is some integer. So, this system has an infinite number of solutions of the form <math> (\pi k, 0) </math>; that is, the solution set for this system of equations is {...<math> (-2\pi, 0) </math>, <math> (-\pi, 0) </math>, <math> (0, 0) </math>, <math> (\pi, 0) </math>, <math> (2\pi, 0) </math>...}.
 ==Resources==

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 Here are some examples of systems of equations with two variables:
-* <math> y = 2x </math> and <math> y = x + 5 </math>. By use of substitution, elimination, or graphing, we can find that the solution to this system is <math> (5, 10) </math>, as this is the only point shared by both equations in the system.
+* One solution: <math> y = 2x </math> and <math> y = x + 5 </math>. By use of substitution, elimination, or graphing, we can find that the solution to this system is <math> (5, 10) </math>, as this is the only point shared by both equations in the system.
-* <math> y = x^2 </math> and <math> y = 4 </math>. Since <math> x^2 = 4 </math> when x = 2 or -2, there are two solutions to this system: (2, 4) and (-2, 4). This is also clear when we graph these two equations, as <math> y = 4 </math> intersects the parabola <math> y = x^2 </math> when x = -2, and when x = 2.
+* Multiple solutions: <math> y = x^2 </math> and <math> y = 4 </math>. Since <math> x^2 = 4 </math> when x = 2 or -2, we know there are two solutions to this system: (2, 4) and (-2, 4). This is also clear when we graph these two equations, as <math> y = 4 </math> intersects the parabola <math> y = x^2 </math> when x = -2, and when x = 2.
 ==Resources==