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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="../539/functions.html">Functions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Equation of a Straight Line</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/functions/straight-line-graphs/main-1.png" style="float: left; width: 100px; height: 100px;"></p> <p>You will probably recognise the most commonly used form of the equation of a straight line: y = mx + c where m represents the gradient and c represents the y-intercept. It is important not to overlook this topic, as it comes up a lot in work on Functions, Vectors and Calculus. Make sure that you can use all three forms of the equation of a straight line, as this will help you with the <a href="../744/equation-of-tangent-and-normal.html" title="Equation of Tangent and Normal">Equation of Tangent and Normal.</a></p> <hr class="hidden-separator"> <p> </p> <p style="text-align: center;"> </p> <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"> <p>On this page, you should learn to</p> <ul> <li>Use the three different forms of the equations of straight lines <ul> <li><span class="math-tex">\(y = mx + c \)</span></li> <li><span class="math-tex">\(ax+by + d = 0 \)</span></li> <li><span class="math-tex">\(y-y_1=m(x-x_1) \)</span></li> </ul> </li> <li>Solve problems with parallel lines <span class="math-tex">\(m_1=m_2\)</span></li> <li>Solve problems with perpendicular lines <span class="math-tex">\(m_1 \times m_2=-1\)</span></li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <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/functions/straight-line-graphs/revision-notes_stline_graphs.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/functions/straight-line-graphs/revision-notes_stline_graphs.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </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="#7bc2db70"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="7bc2db70"> <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%;"> Equation of a Straight Line <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="11-189-1913" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>What is the gradient and y intercept of the following line?</p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q1.jpg" style="width: 300px; height: 441px;"></p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>y intercept = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>gradient = <span class="math-tex">\(\frac{rise}{run}\)</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 equation of the following line can be written in the form <span class="math-tex">\(y=ax+b\)</span></p><p>Find <em><strong>a</strong></em> and <strong><em>b</em></strong></p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q2a.jpg" style="width: 400px; height: 467px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>a = gradient</p><p>b = y intercept</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>Which of the following is the correct equation of the following straight line</p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q3.jpg" style="width: 244px; height: 450px;"></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>y = 3x - 1</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = -3x - 1</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = -x +3</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = 2x + 1</span></label> </p></div><div class="q-explanation"><p>gradient = 3</p><p>y intercept = -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>Which of the following lines passes through the point (2 , 1)</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> y = 2x - 1</label></p><p><label class="radio"><input type="radio"> y = 2x + 1</label></p><p><label class="radio"><input class="c" type="radio"> y = -x + 3</label></p><p><label class="radio"><input type="radio"> y = x - 2</label></p></div><div class="q-explanation"><p>y = -x + 3</p><p>1 = -2 + 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 class="exercise shadow-bottom"><div class="q-question"><p>Which of the following lines is parallel to <span class="math-tex">\(y = \frac{1}{2}x+2\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>x + 2y + 3 = 0</span></label> </p><p><label class="radio"> <input type="radio"> <span>2x - y + 3 = 0</span></label> </p><p><label class="radio"> <input type="radio"> <span>2x + y + 3 = 0</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>x - 2y + 3 = 0</span></label> </p></div><div class="q-explanation"><p>x - 2y + 3 = 0 can be rearranged into the form y = mx + c</p><p>x + 3 = 2y</p><p><span class="math-tex">\(y = \frac{1}{2}x+\frac{3}{2}\)</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>Which of he following gradients could represent the gradients of 2 perpendicular lines</p><p>A) 2</p><p>B) <span class="math-tex">\(\frac{2}{5}\)</span></p><p>C) 5</p><p>D) <span class="math-tex">\(-\frac{5}{2}\)</span></p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B and C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A and D</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A and B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> B and D</label></p></div><div class="q-explanation"><p>If a line has gradient <strong><em>m </em></strong>then a perpendicular line has gradient <span class="math-tex">\(\frac{-1}{m}\)</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>What is the gradient of the line that is <strong>perpendicular</strong> to the line below</p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q1.jpg" style="width: 300px; height: 441px;"></p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></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>The gradient a line <strong>perpendicular </strong>to <span class="math-tex">\(y = -\frac{2}{3}x-2\)</span> can be written in the form <span class="math-tex">\(\frac{a}{b}\)</span></p><p>Find <em><strong>a</strong></em> and <strong><em>b</em></strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>If a line has gradient <strong><em>m </em></strong>then a perpendicular line has gradient <span class="math-tex">\(\frac{-1}{m}\)</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 equation line below can be written in the form <strong><em><span class="math-tex">\(ax+by+c=0\)</span></em></strong></p><p>Find <strong><em>a </em></strong>,<strong><em> b</em></strong> and <strong><em>c</em></strong></p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q7.jpg" style="width: 600px; height: 449px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p>gradient = <span class="math-tex">\(-\frac{2}{5}\)</span></p><p>y intercept = -2</p><p><span class="math-tex">\(y = -\frac{2}{5}x-2\\ 5y=-2x-10\\ 5y+2x+10=0\\ 2x+5y+10=0\)</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 equation of the line that is <strong>perpendicular</strong> to the line below and that passes through the point (1,1) can be written in the form <strong><em><span class="math-tex">\(ax+2y+b=0\)</span></em></strong></p><p>Find <strong><em>a </em></strong>and<strong><em> b</em></strong></p><p><img alt="" src="../../files/differentiation/tangents-and-normals/quiz/q7.jpg" style="width: 600px; height: 449px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>gradient of line in graph = <span class="math-tex">\(-\frac{2}{5}\)</span></p><p>gradient of perpendicular line = <span class="math-tex">\(\frac{5}{2}\)</span></p><p><span class="math-tex">\(y = \frac{5}{2}x+c \)</span></p><p>passes through (1,1)</p><p><span class="math-tex">\(1 = \frac{5}{2}\cdot1+c\\ -\frac{3}{2} = c \\ y = \frac{5}{2}x-\frac{3}{2} \\ 2y=5x-3\\ -5x+2y+3=0 \)</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> </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> <hr class="hidden-separator"> <p>Here is a quiz about parallel and perpendicular lines</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#47059743"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="47059743"> <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%;"> Parallel and Perpendicular Lines <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="11-526-1913" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Match up the gradients with the equations of the straight lines</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">-2</span> <span class="q-text-draggable draggable" draggable="true">-0.5</span> <span class="q-text-draggable draggable" draggable="true">0.5</span> <span class="q-text-draggable draggable" draggable="true">2</span></p><p>y = 2x + 1 , gradient = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>2x + y + 1 = 0 , gradient = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p><span class="math-tex">\(y-1= \frac{1}{2}(x+2)\)</span> , gradient = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p><p>x + 2y - 1 = 0 , gradient = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>y = <strong>m</strong>x + c , gradient = <strong>m</strong></p><hr class="hidden-separator"><p>y = 2x + 1 , gradient = 2</p><hr class="hidden-separator"><p>2x + y + 1 = 0</p><p>y = -2x - 1 , gradient = -2</p><hr class="hidden-separator"><span class="math-tex">\(y-1= \frac{1}{2}(x+2)\)</span> , gradient = 0.5<hr class="hidden-separator"><p>x + 2y - 1 = 0</p><p>y = -0.5x + 0.5 , gradient = -0.5</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 equation of a line that is parallel to y = 3x - 2 and passes through the point (1 , 2) is given by</p><p><strong>y = mx + c</strong></p><p>Find <strong>m</strong> and <strong>c</strong></p></div><div class="q-answer"><p><strong>m</strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p><strong>c</strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p>If the line is parallel to y = 3x - 2 , then the gradient = 3</p><p>We can write it in the form</p><p>y = 3x + c</p><p>We know that it passes through (1 , 2)</p><p>2 = 3(1) + c</p><p>c = -1</p><p>Therefore, the equation is y = 3x - 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 equation of a line that is parallel to <span class="math-tex">\(y=\frac{2}{3}x+1\)</span> and passes through (3 , -1) is given by</p><p><strong>ax + by + d = 0</strong> , where a > 0</p><p>Find <strong>a</strong> , <strong>b </strong>and <strong>d</strong></p></div><div class="q-answer"><p><strong>a</strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong>b</strong> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><p><strong>d</strong> = <input type="text" style="height: auto;" data-c="-9"> <span class="review"></span></p></div><div class="q-explanation"><p>If the line is parallel to <span class="math-tex">\(y=\frac{2}{3}x+1\)</span> , then the gradient = <span class="math-tex">\(\frac{2}{3}\)</span></p><p>We can write it in the form</p><p><span class="math-tex">\(y=\frac{2}{3}x+c\)</span></p><p>We know that it passes through (3 , -1)</p><p>-1 = <span class="math-tex">\(\frac{2}{3}\)</span>(3) + c</p><p>c = -3</p><p>The equation is y = <span class="math-tex">\(\frac{2}{3}\)</span>x - 3</p><p>Multiply through by 3 and rearrange to put in the correct form:</p><p>3y = 2x - 9</p><p>2x - 3y - 9 = 0</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 equation of the line <span class="math-tex">\(l_1\)</span> is given by 2x + 3y - 5 = 0</p><p><span class="math-tex">\(l_2\)</span> is parallel to <span class="math-tex">\(l_1\)</span></p><p>Which of the following could be the equation of the line <span class="math-tex">\(l_2\)</span></p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(y=\frac{2}{3}x+1\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(y=-\frac{3}{2}x+2\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(y = -\frac{2}{3}x\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> 4x + 6y - 1 = 0</label></p></div><div class="q-explanation"><p>2x + 3y - 5 = 0</p><p><span class="math-tex">\(y = -\frac{2}{3}x+\frac{5}{3}\)</span></p><p>Therefore the gradients of the lines <span class="math-tex">\(l_1\)</span> and <span class="math-tex">\(l_2\)</span> are <span class="math-tex">\(-\frac{2}{3}\)</span></p><hr class="hidden-separator"><p>Rearrange 4x + 6y - 1 = 0 to find the gradient</p><p><span class="math-tex">\(y = -\frac{4}{6}x+\frac{1}{6}\)</span></p><p>The gradient of this line is <span class="math-tex">\(-\frac{4}{6}=-\frac{2}{3}\)</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>A straight line, <em><span class="math-tex">\(l\)</span></em> has gradient = <span class="math-tex">\(\frac{1}{2}\)</span></p><p>What is the gradient of a line perpendicular to <em><span class="math-tex">\(l\)</span>?</em></p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>If the gradients of the lines are <span class="math-tex">\(m_1\)</span> and <span class="math-tex">\(m_2\)</span>, then <span class="math-tex">\(m_1 \times m_2=-1\)</span></p><p><span class="math-tex">\(\frac{1}{2} \times m_2=-1\)</span></p><p><span class="math-tex">\(\frac{1}{2} \times (-2)=-1\)</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><img alt="" src="../../files/functions/straight-line-graphs/q6" style="width: 300px; height: 255px;"></p><p>The gradient of <span class="math-tex">\(l_1\)</span> is <span class="math-tex">\(m_1\)</span></p><p>The gradient of <span class="math-tex">\(l_2\)</span> is <span class="math-tex">\(m_2\)</span></p><p>What is <span class="math-tex">\(m_1 \times m_2\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(m_1 \times m_2\)</span> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p>Since these lines are perpendicular, then <span class="math-tex">\(m_1 \times m_2=-1\)</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>What is the gradient of a line that is <strong>perpendicular</strong> to the one in the graph?</p><p><img alt="" src="../../files/functions/straight-line-graphs/q5" style="width: 300px; height: 297px;"></p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/functions/straight-line-graphs/q5ans" style="width: 300px; height: 297px;"></p><p>The gradient of the line in the graph is <span class="math-tex">\(-\frac{2}{3}\)</span></p><p>The gradient of the perpendicular line = <span class="math-tex">\(\frac{-1}{-\frac{2}{3}}=\frac{3}{2}\)</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 equation of the line <span class="math-tex">\(l_1\)</span> is given by</p><p><strong>y - 3 = -2 (x - 2)</strong></p><p><span class="math-tex">\(l_2\)</span> is perpendicular to <span class="math-tex">\(l_1\)</span></p><p>What is the gradient of <span class="math-tex">\(l_2\)</span>?</p></div><div class="q-answer"><p>gradient = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>gradient of <span class="math-tex">\(l_1\)</span>= -2</p><p>gradient of <span class="math-tex">\(l_2=-\frac{1}{-2}=\frac{1}{2}\)</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 equation of the line <span class="math-tex">\(l_1\)</span> is given by 2x + 5y - 5 = 0</p><p><span class="math-tex">\(l_2\)</span> is perpendicuar to <span class="math-tex">\(l_1\)</span></p><p>Which of the following could be the equation of the line <span class="math-tex">\(l_2\)</span></p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(y=-\frac{2}{5}x+1\)</span></label></p><p><label class="checkbox"><input type="checkbox"> 5x + 2y - 5 = 0</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 5x - 2y + 3 = 0</label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(y = \frac{5}{2}x-1\)</span></label></p></div><div class="q-explanation"><p>2x + 5y - 5 = 0</p><p><span class="math-tex">\(y = -\frac{2}{5}x+1\)</span></p><p>The gradient of the line <span class="math-tex">\(l_1\)</span>= <span class="math-tex">\(-\frac{2}{5}\)</span></p><p>Therefore, the gradient of the line <span class="math-tex">\(l_2\)</span>= <span class="math-tex">\(\frac{5}{2}\)</span></p><hr class="hidden-separator"><p>Rearrange 5x - 2y + 3 = 0 to find the gradient</p><p>2y = 5x + 3</p><p><span class="math-tex">\(y=\frac{5}{2}x+\frac{3}{2}\)</span></p><p>The gradient of this line is <span class="math-tex">\(\frac{5}{2}\)</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><img alt="" src="../../files/functions/straight-line-graphs/q10" style="width: 300px; height: 297px;"></p><p>The lines <span class="math-tex">\(l_1,l_2, l_3 \)</span> and <span class="math-tex">\( l_4\)</span> form a square.</p><p><span class="math-tex">\( l_1:\)</span> x + 2y = 12</p><p><span class="math-tex">\( l_2:\)</span> 2x - y = -6</p><p><span class="math-tex">\( l_3: \ y=-\frac{1}{2}x+\frac{7}{2}\)</span></p><p>Which of the following lines completes the square?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> x - 2y + 1 = 0</label></p><p><label class="checkbox"><input type="checkbox"> 2x + y + 11 = 0</label></p><p><label class="checkbox"><input class="c" type="checkbox"> y = 2x + 1</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 2x - y + 11 = 0</label></p></div><div class="q-explanation"><p>gradient of <span class="math-tex">\( l_1\)</span>= <span class="math-tex">\(-\frac{1}{2}\)</span></p><p>gradient of <span class="math-tex">\( l_2\)</span>= 2</p><p>gradient of <span class="math-tex">\( l_3\)</span>= <span class="math-tex">\(-\frac{1}{2}\)</span></p><p>Therefore, gradient of <span class="math-tex">\( l_4\)</span>= 2</p><p>There are two possible solutions:</p><p><img alt="" src="../../files/functions/straight-line-graphs/q10ans1" style="width: 300px; height: 280px;"> <img alt="" src="../../files/functions/straight-line-graphs/q10ans2" style="width: 300px; height: 280px;"></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="874"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>The line <span class="math-tex">\(l_1\)</span> has equation 2x+5y+6=0</p> <p>The line <span class="math-tex">\(l_2\)</span> is perpendicular to the line <span class="math-tex">\(l_1\)</span>and passes through the point (2 , -2)</p> <p>a) Find the equation of <span class="math-tex">\(l_2\)</span>in the form ax+by+d=0, where a, b and d are constants.</p> </div> <p>b) Find the coordinates where <span class="math-tex">\(l_2\)</span> meets the y axis</p> <hr class="hidden-separator"> <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"><content> </content>If the gradient of a line is <em><strong>m</strong></em> , then the gradient of a perpendicular line is <span class="math-tex">\({-1 \over m}\)</span></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/functions/straight-line-graphs/esq_functions_stlines1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/straight-line-graphs/esq_functions_stlines1.pdf" width="640"></iframe></p> </section> </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="872"> <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/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL 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>The point A has coordinates (a , 3) and the point B has coordinates (7 , b).</p> <p>The line AB has equation 2x + 3y = 11.</p> <p>a) Find the values of a and b</p> <p>The line AC is perpendicular to the line AB.</p> <p>b) Find the equation of the line AC in the form ax + by + d = 0, where a, b and d are constants</p> <p>c) Given that C lies on the x axis, find its coordinates.</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">b) the line AC is perpendicular to AB and passes through the point A (found in part a)<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/functions/straight-line-graphs/esq_functions_stlines2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/straight-line-graphs/esq_functions_stlines2.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="873"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The line <span class="math-tex">\(l_1 \)</span> passes through the point P(3k , 2k) with gradient = -2.</p> <p><span class="math-tex">\(l_1 \)</span> meets the x axis at A and the y axis at B.</p> <p>a) Find the equation of the line <span class="math-tex">\(l_1 \)</span> and show that A(4k , 0)</p> <p>b) Find the area of the triangle AOB in terms of k</p> <p>The line <span class="math-tex">\(l_2\)</span> passes through P and is perpendicular to <span class="math-tex">\(l_1 \)</span></p> <p>c) Find the equation of <span class="math-tex">\(l_2\)</span></p> <p><span class="math-tex">\(l_2\)</span>meets the x axis at C</p> <p>d) Show that the midpoint of PC lies on the line y = x</p> <hr class="hidden-separator"> <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>The key to this type of question is to be able to visualize it.</p> <p><content>Draw a sketch to help you:</content></p> <p><img alt="" src="../../files/functions/straight-line-graphs/esq3-hint.png" style="width: 400px; height: 353px;"></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/functions/straight-line-graphs/esq_functions_stlines3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/straight-line-graphs/esq_functions_stlines3.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="871"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Point A has coordinates (<em><strong>a</strong></em> , 6) and point B has coordinates (5 , <em><strong>b</strong></em>).</p> <p>The line 8x - 6y + 3 = 0 is the perpendicular bisector of AB.</p> <p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong>.</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>Find the gradient of AB in terms of <strong><em>a</em></strong> and <strong><em>b</em></strong>.</p> <p><content>Gradient of AB is perpendicular to line 8x - 6y + 3 = 0 </content></p> <hr> <p> </p> <p><content>Find the midpoint of AB</content></p> <p>Midpoint should lie on line 8x - 6y + 3 = 0</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/functions/straight-line-graphs/esq_functions_stlines4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/straight-line-graphs/esq_functions_stlines4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div 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