{"id":206,"date":"2017-08-23T09:59:28","date_gmt":"2017-08-23T09:59:28","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=206"},"modified":"2017-10-20T06:05:18","modified_gmt":"2017-10-20T06:05:18","slug":"geometric-interpretations","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/geometric-interpretations\/","title":{"rendered":"Geometric Interpretations"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/matrices-for-systems-of-equations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/special-matrix-products\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Geometric Interpretations<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will compare symbolic and geometric interpretations of matrices. You will also infer geometric\u00a0interpretations of the determinants of and matrices.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>A linear equation of the form\u00a0<em>ax<\/em> + <em>by<\/em> = <em>c<\/em> is a line when graphed on\u00a0coordinate axes.<\/li>\n<li>A linear equation of the form <em>ax<\/em> + <em>by<\/em> + <em>cz\u00a0<\/em>= <em>d<\/em> is a plane when graphed on coordinate axes.<\/li>\n<\/ul>\n<section>\n<h3>How does the solution of a system of equations relate to\u00a0graphing?<\/h3>\n<p>Consider a system of two-variable linear equations:<\/p>\n<ul>\n<li>there may be one solution.<\/li>\n<li>there may be no solution.<\/li>\n<li>there may be an infinite number of solutions.<\/li>\n<\/ul>\n<p>Similarly, in a system of three-variable linear equations:<\/p>\n<ul>\n<li>there may be one solution.<\/li>\n<li>there may be no solution.<\/li>\n<li>there may be an infinite number of solutions.<\/li>\n<\/ul>\n<p>Below is a system of two-variable linear equations with one\u00a0solution.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.4%20Art%20001.JPG\" alt=\"System of 2 variable linear equations\" width=\"89\" height=\"59\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>When a system of two-variable equations has a solution, each\u00a0equation can be graphed as a line. The solution set is the\u00a0coordinates of the point where the two lines cross.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.4%20Art%20002.JPG\" alt=\" Graph of system\" width=\"224\" height=\"233\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p class=\"white_lesson_header\" align=\"CENTER\">Solving the System<\/p>\n<p align=\"CENTER\">By Algebraic Symbols<\/p>\n<p align=\"CENTER\">By Matrices<\/p>\n<p align=\"CENTER\"><em>y<\/em> \u2013 2<em>x<\/em> = 4<\/p>\n<p align=\"CENTER\">\u2013 (<em>y <\/em>+ <em>x<\/em> = \u2013 5 )<\/p>\n<p align=\"CENTER\">\u2013 3<em>x<\/em> = 9<\/p>\n<p align=\"CENTER\"><em>x<\/em> = \u2013 3<\/p>\n<p align=\"CENTER\"><em>y <\/em>\u2013 2(\u2013 3) = 4<\/p>\n<p align=\"CENTER\"><em>y <\/em>= \u2013 2<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p2_clip_image003.gif\" width=\"97\" height=\"291\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Solving the system results in the point (\u20133, \u20132).<\/p>\n<p>If a system of equations has no solution, it is called\u00a0<abbr title=\"A system of equations is inconsistent if there is no solution to the system. \">inconsistent<\/abbr>.<\/p>\n<p>For example,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p2_clip_image006.gif\" width=\"96\" height=\"20\" name=\"graphics6\" align=\"TEXTTOP\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p2_clip_image009.gif\" width=\"87\" height=\"43\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.4%20Art%20003.JPG\" alt=\"Graph of inconsistent equations\" width=\"224\" height=\"233\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The two lines defined by the equations are <abbr title=\"extending in the same direction, everywhere equidistant,coplanar, and not meeting \">parallel<\/abbr>.\u00a0Because they never meet, there is no solution to the system of\u00a0equations. An inconsistent system can be recognized by\u00a0manipulating the system, and finding the result to be an\u00a0inconsistent statement. In this case, the solution process yields\u00a0the inconsistent statement<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s7_p2_html_2f9e4515.gif\" width=\"157\" height=\"21\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p class=\"white_lesson_header\" align=\"CENTER\">Solving the System<\/p>\n<p align=\"CENTER\">By Algebraic Symbols<\/p>\n<p align=\"CENTER\">By Matrices<\/p>\n<p align=\"CENTER\">2<em>x<\/em> + 3<em>y<\/em> = \u2013 3<\/p>\n<p align=\"CENTER\"><em>y<\/em> =<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p2_clip_image012.gif\" width=\"55\" height=\"33\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\">2<em>x<\/em> + 3<em>y<\/em> = \u2013 3<\/p>\n<p align=\"CENTER\">2<em>x<\/em> + 3<em>y<\/em> = 6<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p2_html_m4f3b135a.gif\" width=\"87\" height=\"21\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\">\u2013 (2<em>x<\/em> + 3<em>y<\/em> = 6)<\/p>\n<p align=\"CENTER\">0 = \u20139<\/p>\n<p align=\"CENTER\">This system is called inconsistent because both\u00a0equations cannot be true at the same time.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p2_clip_image015.gif\" width=\"91\" height=\"144\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\">Notice that an inconsistent system in two\u00a0variables indicates two lines that are parallel since the bottom\u00a0row doesn&#8217;t have a one in it.<\/p>\n<p class=\"lesson_subhead\"><strong>How does a system of two-variable linear equations with\u00a0an infinite number of solutions relate to graphing?<\/strong><\/p>\n<p>If a system of equations has an infinite number of solutions,\u00a0it is said to be <abbr title=\"when one equation can be shown to be equivalent to another \">redundant<\/abbr>.<\/p>\n<p align=\"CENTER\">5<em>x<\/em> \u2013 2<em>y<\/em> = 7<\/p>\n<p align=\"CENTER\">15<em>x<\/em> \u2013 6<em>y <\/em>= 21<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.4%20Art%20004.JPG\" alt=\"Redundant Graph\" width=\"224\" height=\"233\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The graph of the line is purple in order to indicate that the\u00a0red and the blue lines are the same. This system is redundant\u00a0because both equations yield the same information\u2014the same\u00a0points. There are an infinite number of solutions to this system.<\/p>\n<p>The two lines defined by the equations are the same line.\u00a0Because they meet everywhere, there are an infinite number of\u00a0solutions to this system of equations. A redundant system can be\u00a0recognized symbolically by manipulating the system, and finding\u00a0the result to be a universally true statement. In this case, the\u00a0solution process yields the universally true statement<\/p>\n<p align=\"CENTER\">0 = 0, or 0<em>x<\/em> + 0<em>y<\/em> = 0.<\/p>\n<p class=\"white_lesson_header\" align=\"CENTER\">Solving the System<\/p>\n<p align=\"CENTER\">By Algebraic Symbols<\/p>\n<p align=\"CENTER\">5<em>x<\/em> \u2013 2<em>y<\/em> = 7<\/p>\n<p align=\"CENTER\">15<em>x<\/em> \u2013 6<em>y<\/em> = 21<\/p>\n<p align=\"CENTER\">\u20133(5<em>x<\/em> \u2013 2<em>y<\/em> = 7)<\/p>\n<p align=\"CENTER\">15<em>x<\/em> \u2013 6<em>y<\/em> = 21<\/p>\n<p align=\"CENTER\">0 = 0<\/p>\n<p>This system is called redundant because both equations yield\u00a0the same information.<\/p>\n<p>The solution can be characterized as<\/p>\n<p><em>x<\/em> = any real number and <em>y<\/em> =<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s7_p3_html_m607a11f6.gif\" width=\"51\" height=\"41\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p3_clip_image006.gif\" width=\"99\" height=\"123\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Notice that a redundant system in two variables indicates both\u00a0lines are the same\u2014that both equations describe the same\u00a0relationship.<\/p>\n<h3>How do we graph a system of three-variable linear equations\u00a0with one solution?<\/h3>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p4_clip_image003.gif\" width=\"109\" height=\"68\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>In this system, the three planes described by the equations\u00a0meet at one point. (Think about the corner of a room or the vertex\u00a0of a triangular pyramid.) Because there is one solution, the graph\u00a0of the solution is a point.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"white_lesson_header\" align=\"CENTER\">Solution of System by Using Matrices<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p4_clip_image006.gif\" width=\"129\" height=\"316\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p align=\"CENTER\">Go to column 2.<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p4_clip_image009.gif\" width=\"123\" height=\"197\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p4_clip_image012.gif\" width=\"123\" height=\"117\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p align=\"CENTER\">Go to column 3.<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p4_clip_image015.gif\" width=\"121\" height=\"272\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p align=\"CENTER\">Solution:<\/p>\n<p align=\"CENTER\"><em>x<\/em> = 4<\/p>\n<p align=\"CENTER\"><em>y<\/em> = \u2013 7<\/p>\n<p align=\"CENTER\"><em>z<\/em> = 11<\/p>\n<p>How do you graph a system of three-variable linear equations\u00a0with no solution?<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p4_clip_image018.gif\" width=\"109\" height=\"68\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>This system has <em><strong>no solution<\/strong><\/em>.\u00a0Therefore, the graph will show three planes that never meet at the\u00a0same place. For example, the three walls of a triangular room meet\u00a0in pairs, but all three planes never meet simultaneously. This\u00a0system is inconsistent.<\/p>\n<h3>Finally, how do you graph a system of three-variable linear\u00a0equations with an infinite number of solutions?<\/h3>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p5_clip_image003.gif\" width=\"109\" height=\"69\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>This system has an infinite number of solutions. Therefore, the\u00a0graph of the solution is the line where two planes meet. This\u00a0system is redundant.<\/p>\n<p class=\"white_lesson_header\" align=\"CENTER\">Solution of System by Matrices<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p5_clip_image006.gif\" width=\"131\" height=\"371\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p style=\"text-decoration: none;\">The final matrix has a row of all\u00a0zeros. From elementary linear algebra we know that we need at\u00a0least 3 equations to solve for 3 variables. Each row in the matrix\u00a0corresponds to one of our equations, and the row of all zeros\u00a0corresponds to an equation from which we get no information.\u00a0Therefore, we only have two potentially useful equations, and we\u00a0cannot solve for 3 variables with only two equations. As long as\u00a0the remaining two rows are not inconsistent (in this case they are\u00a0not, but we need to check!), the entire system is redundant. If\u00a0the remaining two rows were inconsistent, the entire system would\u00a0be inconsistent.<\/p>\n<p><span style=\"text-decoration: none;\">To represent the\u00a0solution set, let <\/span><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">be any real number, and solve for the other two variables in terms\u00a0of <em>z<\/em>. <\/span><\/p>\n<p style=\"text-decoration: none;\"><em>z<\/em> = any real number.<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\">\u20137<em>y<\/em> \u2013<br \/>\n11 z = \u20139<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p5_clip_image009.gif\" width=\"96\" height=\"41\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><em>x<\/em> \u2013 3<em>y<br \/>\n<\/em>\u2013 5<em>z<\/em> = \u2013 3<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s7_p5_html_m7517d52f.gif\" width=\"91\" height=\"41\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\">Thus, if <em>z<\/em> is\u00a0arbitrarily chosen as \u2013 3, then\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s7_p5_html_m4b8c9143.gif\" width=\"47\" height=\"41\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/>\u00a0and <em>y<\/em> = 6.<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\">Or if <em>z<\/em> is arbitrarily chosen as 4,\u00a0then\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s7_p5_html_5ce0838a.gif\" width=\"49\" height=\"41\" name=\"graphics10\" align=\"absmiddle\" border=\"0\" \/>\u00a0and <em>y<\/em> = \u20135.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"4\">Geometric Interpretations<\/th>\n<\/tr>\n<tr>\n<th width=\"15%\">\n<div align=\"center\">Number of Variables in the Linear System<\/div>\n<\/th>\n<th width=\"27%\">\n<div align=\"center\">One Solution<\/div>\n<\/th>\n<th width=\"29%\">\n<div align=\"center\">No Solution<\/div>\n<\/th>\n<th width=\"29%\">\n<div align=\"center\">Infinite Number of Solutions<\/div>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\n<div align=\"center\"><strong><span style=\"text-decoration: none;\">2 <\/span><\/strong><\/div>\n<\/td>\n<td>\n<div align=\"center\">Equations meet at a point. Graph shows two lines<br \/>\nthat cross.<\/div>\n<\/td>\n<td>\n<div align=\"center\">Equations do not meet. Graph shows two parallel<br \/>\nlines.<\/div>\n<\/td>\n<td>\n<div align=\"center\">Equations yield the same relationship. Graph shows<br \/>\nthe two lines coincide.<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\"><strong><span style=\"text-decoration: none;\">3 <\/span><\/strong><\/div>\n<\/td>\n<td>\n<div align=\"center\">Equations meet at a point. Graph shows three<br \/>\nplanes that meet in one location.<\/div>\n<\/td>\n<td>\n<div align=\"center\">Equations do not all three meet. Graph shows<br \/>\nplanes that do not meet simultaneously.<\/div>\n<\/td>\n<td>\n<div align=\"center\">At least one pair of equations is redundant. Graph<br \/>\nshows at least two planes that meet at a line.<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>If the following is a reduced matrix form of a system of linear\u00a0equations, how many solutions are there?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p6_clip_image003.gif\" width=\"113\" height=\"73\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>0<\/li>\n<li>1<\/li>\n<li>2<\/li>\n<li>An infinite number<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct choice is A.\u00a0The final row indicates that 0<\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">+ 0<\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">+ 0<\/span><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 9, which is impossible. Thus there are no solutions<\/span>.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>If the following is a matrix form of a system of linear\u00a0equations, what does the geometric representation look like?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p6_clip_image006.gif\" width=\"121\" height=\"73\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>Three lines that meet at the same point<\/li>\n<li>Three planes that meet at the same point<\/li>\n<li>Three planes that never meet<\/li>\n<li>Two planes that meet on a line<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B. Rewriting the matrix in reduced-row\u00a0echelon form yields:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p6_clip_image008.gif\" width=\"102\" height=\"70\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Therefore, the common solution for the system of equations\u00a0is (x,y,z)=(-1,-3,0). That coordinate, alone, is shared by all\u00a0three equations. Note that equations containing three unknowns\u00a0indicate three-dimensional figures, in this case planes.<\/p>\n<\/div>\n<\/section>\n<h3>Determinants of Matrices Also Have a Geometric Interpretation.<\/h3>\n<p>Suppose you have the matrix below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p7_clip_image003.gif\" width=\"51\" height=\"49\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This combination can be viewed as the two vectors [3, 0] and [1, 1].<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.4%20Art%20005.JPG\" alt=\" Combination of the two vectors\" width=\"189\" height=\"165\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Notice that the two vectors describe a parallelogram.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.4%20Art%20006.JPG\" alt=\"Two vectors describing a parallelogram\" width=\"188\" height=\"164\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This parallelogram has a height of 1 and a width of 3. Thus its\u00a0area is 1(3) = 3.<\/p>\n<p>Notice that the determinant of our matrix is 3(1) \u2013 1(0)\u00a0= 3.<\/p>\n<h3>What about triangles?<\/h3>\n<p>Suppose you have a triangle with vertices at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image003.gif\" width=\"149\" height=\"21\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0The area of the triangle can be determined by finding a\u00a0determinant. The area is the absolute value of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image006.gif\" width=\"87\" height=\"73\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Note: the area must be positive, thus the absolute value of the\u00a0expression is imperative.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the area of a triangle with vertices of (3, 1), (4,\u00a0\u20137), and (0, \u20131)?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image009.gif\" width=\"31\" height=\"23\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>\u00a0square units<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image012.gif\" width=\"33\" height=\"23\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>\u00a0square units<\/li>\n<li>13 square units<\/li>\n<li>26 square units<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is C. The solution is the absolute value of<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image015.gif\" width=\"595\" height=\"73\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<p>Can determinants identify colinearity?<\/p>\n<p>It follows from the triangle relation that if and only if three\u00a0points\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image018.gif\" width=\"149\" height=\"21\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are collinear, then<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image021.gif\" width=\"100\" height=\"73\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which is a value for <em>x<\/em> such that the three points (<em>x<\/em>,\u00a01), (8, 4), and (\u20132, <em>x \u2013 <\/em>5) are collinear?<\/p>\n<ol>\n<li><em>x<\/em> = \u201312<\/li>\n<li><em>x<\/em> = \u20133<\/li>\n<li><em>x<\/em> = 5<\/li>\n<li><em>x<\/em> = 14<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. There are two solutions to this\u00a0problem: 3 and 14. Using the determinant method,<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image024.gif\" width=\"569\" height=\"73\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The quadratic equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image027.gif\" width=\"116\" height=\"20\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>\u00a0has solutions of 3 and 14.<\/p>\n<\/div>\n<\/section>\n<p>Can the equation of a line be found with determinants?<\/p>\n<p>A line can be identified using two points\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image030.gif\" width=\"93\" height=\"21\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Using a generic point (<em>x<\/em>, <em>y<\/em>), the equation of a<br \/>\nline can be found using the following determinant equation.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s7_p8_clip_image033.gif\" width=\"100\" height=\"73\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/matrices-for-systems-of-equations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/special-matrix-products\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Geometric Interpretations Objective In this lesson, you will compare symbolic and geometric interpretations of matrices. You will also infer geometric\u00a0interpretations of the determinants of and matrices. Previously Covered: A linear equation of the form\u00a0ax + by = c is a line when graphed on\u00a0coordinate axes. A linear equation of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-206","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/206","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=206"}],"version-history":[{"count":12,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/206\/revisions"}],"predecessor-version":[{"id":840,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/206\/revisions\/840"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}