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onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../540/geometry-trigonometry.html">Geometry & Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Intersection of Planes</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/vectors/intersection-of-planes/main.jpg" style="float: left; width: 100px; height: 100px;">This page is all about how planes meet (or not). Two planes will either be parallel or meet at a line. Three planes will either 1) not meet - zero solutions, 2) meet at a point - unique solution, or 3) meet at a line (or plane) - infinite solutions. We will look at how to recognise each situation and how to find the solutions. This work overlaps with finding solutions to systems of linear equations.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about</p> <ul> <li>the intersection of two planes</li> <li>the intersection of three planes (unique, infinite and zero solutions)</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>The following videos will help you understand all the concepts from this page</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Unique Solution</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="368"> <p>In this video, we see that systems of equations with 3 unknowns can be represented in 3D space by planes. We use the method of elimination to solve the following system and find the intersection of the 3 planes.</p> <p style="text-align: center;">x + y + 2z = 0</p> <p style="text-align: center;">2x - y + z = -6</p> <p style="text-align: center;">3x + 4y - z = -6</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/249073257"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/intersection-of-planes/systems-of-equations---unique-solution.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/systems-of-equations---unique-solution.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Zero or Infinite Solutions</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="369"> <p>Previously we have seen what happens when there is a unique solution to a system of equations. Here we will look at what happens when we do <strong>not</strong> get a unique solution. In other words, we will consider when there is no solution (zero) and when there are infinite solutions. As well as solving the system of equations algebraically using the elimination method, we will look at what happens graphically: how the three planes that meet (or not).</p> <p>The two systems of equations that we will use to illustrate this are</p> <p style="text-align: center;">x + 3y - 2z = 7</p> <p style="text-align: center;">2x - 2y + z = 3</p> <p style="text-align: center;">3x + y - z = 12</p> <p>and</p> <p style="text-align: center;">x + 3y - 2z = 7</p> <p style="text-align: center;">2x - 2y + z = 3</p> <p style="text-align: center;">3x + y - z = 10</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/249227420"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/intersection-of-planes/systems-of-equations---zero-or-infinite-solutions.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/systems-of-equations---zero-or-infinite-solutions.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div>2 Planes</div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="370"> <p>When we consider 2 planes, they can either</p> <ol> <li>be parallel (an therefore never meet)</li> <li>meet at a line</li> <li>be coincident</li> </ol> <p>In the following video, we are going to look at the second case and see how to find the equation of the line where two planes meet.The method that we will use here gives the solution just like your calculator would.</p> <p>We will illustrate this by solving the following system of equations:</p> <p style="text-align: center;">x + 3y - 2z = 7</p> <p style="text-align: center;">2x - 2y + z = 3</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/249361081"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/intersection-of-planes/intersection-of-2-planes---finding-equation-of-line.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/intersection-of-2-planes---finding-equation-of-line.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div>3 Planes - Infinite Solutions</div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="371"> <p>In the following example, we are going to solve a system of equations. This system has infinite solutions and we will find the general solution. We will see that the 3 equations represent 3 planes and the planes meet at a line. The method is the same as finding the intersection of 2 planes.</p> <p>Show that the following system of equations has infinite solutions and find the general solution of this system</p> <p style="text-align: center;">x + 3y - 2z = 7</p> <p style="text-align: center;">2x - 2y + z = 3</p> <p style="text-align: center;">3x + y - z = 10</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/249521634"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/intersection-of-planes/infinite-solution---finding-equation-of-a-line.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/infinite-solution---finding-equation-of-a-line.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div>Problem Solving</div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="372"> <p>This is a typical exam-style question. It requires you to have a good understanding of what happens when you try to solve a system of equations when there is <strong>infinite solutions</strong> and when there is a <strong>unique solution</strong>.</p> <p>When we eliminate one variable…</p> <ul> <li>Unique Solution - it works, we can solve it!</li> <li>Zero Solution - 2 inconsistent equations</li> <li>Infinite solutions - 2 identical equations</li> </ul> <hr> <p>Find the value(s) of k for which the system of equations below has</p> <ol style="list-style-type:lower-alpha;"> <li>Infinitely many solutions</li> <li>A unique solution</li> </ol> <p style="text-align: center;">2x - y + z = 0</p> <p style="text-align: center;">3x + 2y - z = 8</p> <p style="text-align: center;">x + <strong>k</strong>y + 3z = -<strong>k</strong>² + 8</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/249518786"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/intersection-of-planes/intersection-of-planes---problem-solving.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/intersection-of-planes---problem-solving.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/revision-notes_intersection_of_planes.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/vectors/intersection-of-planes/revision-notes_intersection_of_planes.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#2798b5f6"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="2798b5f6"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Intersection of planes <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-199-655" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Each of the following graphs represents intersecting planes with either a unique, zero or an infinite number of solutions.</p><p>Drag the correct answer into the space provided.</p><p> <span class="q-text-draggable draggable" draggable="true">unique</span> <span class="q-text-draggable draggable" draggable="true">zero</span> <span class="q-text-draggable draggable" draggable="true">infinite</span></p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p><img alt="" src="../../files/vectors/intersection-of-planes/1.jpg" style="width: 350px; height: 350px;"></p><p> <input type="text" style="height: auto;" data-c="unique"> <span class="review"></span></p></td><td><p><img alt="" src="../../files/vectors/intersection-of-planes/2.jpg" style="width: 350px; height: 350px;"></p><p> <input type="text" style="height: auto;" data-c="infinite"> <span class="review"></span></p></td></tr><tr><td><p><img alt="" src="../../files/vectors/intersection-of-planes/3.jpg" style="width: 350px; height: 350px;"></p><p> <input type="text" style="height: auto;" data-c="zero"> <span class="review"></span></p></td><td><p><img alt="" src="../../files/vectors/intersection-of-planes/4.jpg" style="width: 350px; height: 350px;"></p><p> <input type="text" style="height: auto;" data-c="zero"> <span class="review"></span></p></td></tr></tbody></table></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Each of the following systems has a unique, zero or an infinite number of solutions.</p><p>Drag the correct answer into the space provided.</p><p><span class="q-text-draggable draggable" draggable="true">zero</span> <span class="q-text-draggable draggable" draggable="true">infinite</span> <span class="q-text-draggable draggable" draggable="true">unique</span></p></div><div class="q-answer"><p><span class="math-tex">\(x -2y=5\\ 2x-4y=7\)</span> <input type="text" style="height: auto;" data-c="zero"> <span class="review"></span></p><hr><p><span class="math-tex">\(x + y=6\\ 3x+3y=18\)</span> <input type="text" style="height: auto;" data-c="infinite"> <span class="review"></span></p><hr><p><span class="math-tex">\(x + 2y=5\\ 2x+y=4\)</span> <input type="text" style="height: auto;" data-c="unique"> <span class="review"></span></p><hr><p><span class="math-tex">\(2x -3y=5\\ -6x+9y=15\)</span> <input type="text" style="height: auto;" data-c="zero"> <span class="review"></span></p></div><div class="q-explanation"><p>We can re-write each of the systems and it becomes clear</p><p><span class="math-tex">\(2x -4y=10\\ 2x-4y=7 \)</span> equations are inconsistent => zero solutions</p><hr><p><span class="math-tex">\(3x + 3y=18\\ 3x+3y=18\)</span> equations are identital => infinite solutions</p><hr><p><span class="math-tex">\(x + 2y=5\\ 2x+y=4\)</span> equations are not parallel => unique solution</p><hr><p><span class="math-tex">\(-6x+9y=-15\\ -6x+9y=15\)</span> equations are inconsistent => zero solutions</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The point of intersection of the following three planes is (a,b,c). Find the coordinates using your calculator,</p><p><span class="math-tex">\(\ \ x+2y-\ z=5\\ 2x-\ y+3z=-4\\ \ \ x-\ y-\ \ z=2\\\)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="1.2"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="-1.8"> <span class="review"></span></p></div><div class="q-explanation"><p>Use the simulataneous equation solver on your calculator</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the intersection of the plane x+2y-3z=8 with the OXY plane</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> x + 2y = 8</label></p><p><label class="radio"><input type="radio"> 2x - 3y = 8</label></p><p><label class="radio"><input type="radio"> x - 3z = 8</label></p><p><label class="radio"><input type="radio"> z = <span class="math-tex">\(-\frac{8}{3}\)</span></label></p></div><div class="q-explanation"><p>The OXY plane is the plane where z=0</p><p>x+2y-3z=8</p><p>x+2y-3(0)=8</p><p>x+2y=8</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Three planes intersect at a point</p><p><span class="math-tex">\(2x+3y-\ z=0\\ \\ x-\ \ y-\ \ z=2\\ 3x-2y+2z=10\)</span></p><p>Fill in the blanks to complete the solution below</p></div><div class="q-answer"><p>A: 2x + 3y - z = 0</p><p>B: x - y - z = 2</p><p>C: 3x - 2y + 2z = 10</p><hr><p>A-B: x + 4 y = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p>2A+C: 7x + 4y = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> </p><p>6x = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span></p><p>x = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>y = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p>z = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>x + 4y = -2</p><p>7x + 4y = 10</p><p>6x = 12</p><p>x = 2</p><p>Substitute into</p><p>x + 4y = -2</p><p>2 + 4y = -2</p><p>y = -1</p><p>Substitute into</p><p>x - y - z = 2</p><p>2 - (-1) - z = 2</p><p>z = 1</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following system of equations has infinite solutions. Find <strong><em>a </em></strong> and <strong><em>b</em></strong>.</p><p><span class="math-tex">\(\textbf{a}x+3y=\textbf{b}\\ 4x+6y=30\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\((\textbf{2}x+3y=\textbf{15} )\times2\\ 4x+6y=30\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following system of equations has no solution. Find <strong><em>a</em></strong>.</p><p><span class="math-tex">\(15x-5y=8\\ \textbf{a}x+2y=5\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\((15x-5y=8)\times2 \ \ \ \quad => \quad 30x-10y=16\\ (\textbf{-6}x+2y=5)\times-5\quad => \quad 30x-10y=-25\\ \)</span> equations are inconsistent => zero solutions</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following system of equations has no solution. Find <strong><em>k</em></strong> given that <strong><em>k</em></strong><0</p><p><span class="math-tex">\(2kx-9y=10\\ 8x-ky=7\)</span></p></div><div class="q-answer"><p><strong><em>k</em></strong> = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\((2kx-9y=10)\times4\quad => 8kx-36y=40\\ (8x-ky=7)\times k \quad \quad =>8kx-k²y=7k\\ k²=36\\ k=-6 (k<0) \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following system of equations has infinite solutions. Find <strong><em>a </em></strong> and <strong><em>b</em></strong> given that <strong><em>a</em></strong>>0</p><p><span class="math-tex">\(2x-(a+1)y=b\\ ax-\quad \quad10y =30\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\((2x-(a+1)y=b)\times a \ \ \quad => 2ax-a(a+1)y=ba \\ (ax-\quad \quad10y =30)\times 2 \quad => 2ax- \quad \quad 20y=60\\ a(a+1)=20\\ a^2+a-20=0\\ (a+5)(a-4)=0\\ a=4 (a>0)\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The following system of equations is inconsistent. Fill in the blanks and find the value of <strong><em>k</em></strong></p><p><span class="math-tex">\(\ x + 3y - 2z = k² + 1\\ 2x- 2y + \ z = 4\\ kx + \ y - \ z = 6\)</span></p></div><div class="q-answer"><p>A: x + 3y - 2z = k² + 1</p><p>B: 2x - 2y + z = 4</p><p>C: kx + y - z = 6</p><hr><p>B+ C: (2+k)x - y = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> </p><p>A + 2B: 5x - y = k² + <input type="text" style="height: auto;" data-c=" 9"> <span class="review"></span> </p><p>The equations are inconsistent when k = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>B+ C: (2+k)x - y = 10</p><p>A + 2B: 5x - y = k² + 9</p><p>(2 + k) = 5</p><p>k = 3</p><p>If we substitute this value into the equations:</p><p>B+ C: 5x - y = <strong>10</strong></p><p>A + 2B: 5x - y = <strong>18</strong></p><p>This is not possible, so the equations are inconsistent when k = 3</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="448"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the intersection of the planes <span class="math-tex">\({ \Pi }_{ 1 } \)</span> and <span class="math-tex">\({ \Pi }_{ 2 }\)</span> in the form <span class="math-tex">\(\textbf {r}=\textbf {a}+\lambda\textbf {b}\)</span> where the components of <strong><em>b</em></strong> are integers</p> <p style="margin-left: 80px;"><span class="math-tex">\({ \Pi }_{ 1 }:\quad \quad x+2y−z=5\\ { \Pi }_{ 2 }:\quad \quad 2x−y+3z=−4\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>1. Eliminate z and write y in terms of x</p> <p><content>2. Eliminate y and write z in terms of x</content></p> <p>3. Write parametric equations</p> <p style="margin-left: 120px;">x = ...</p> <p style="margin-left: 120px;">y = ...</p> <p style="margin-left: 120px;">z = ...</p> <p>4. Convert to vector form</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes1a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes1a.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="449"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p style="margin-left: 80px;"><span class="math-tex">\(\ \ \ x + \ \ y +\ \ z =8\\ \ \ ax − \ y \ \ \quad \ \ \ =3\\ −x+3y+4z=b\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>There is no unique solution solution to the system of equations. Find the value of <em><strong>a</strong></em>.</li> <li>Given that the system can be solved, find the value of <em><strong>b</strong></em>.</li> </ol> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>1. Eliminate z by combining equation 1 and 3</p> <p>2. Equate the coefficients of y with equation from above and equation 2</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="450"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The three planes <span class="math-tex">\({ \Pi }_{ 1 }\)</span> , <span class="math-tex">\({ \Pi }_{ 2 }\)</span> and <span class="math-tex">\({ \Pi }_{ 3 }\)</span> meet at a line</p> <p><span class="math-tex">\({ \Pi }_{ 1 }:\quad 2x+\ y+3z=a\\ { \Pi }_{ 2 }:\quad \ x−2y+2z=−9 \\ { \Pi }_{ 3 }:\quad 3x+4y+4z=−1\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find a</li> <li>Find the equation of the straight line in the form <span class="math-tex">\(\textbf {r}=\textbf {a}+\lambda\textbf {b}\)</span> where the components of <strong><em>b</em></strong> are integers</li> </ol> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <ul> <li>Combine the equations to eliminate x</li> <li>Equate the coefficients of y and z</li> </ul> <content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="451"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the value of k which makes the following system of equations inconsistent:</p> <p style="margin-left: 80px;"><span class="math-tex">\(x +2y+kz=−1\\ 2x+ \ y− \ z=3\\ kx−2y+ z=1\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Combine the equations to eliminate <strong><em>y</em></strong></p> <p><content>Equate coefficients of <strong><em>z</em></strong></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="655"> <div class="panel-heading">MY PROGRESS</div> <div 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