// Numbas version: exam_results_page_options
{"name": "Equations of parallel lines", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<h4>a)</h4>\n<p>Parallel lines are&nbsp;defined as lines that remain<strong> the same</strong> distance apart and so they never meet. This means that their&nbsp;<strong>gradient</strong> is the same.</p>\n<p>If we write our line equations in the form $y_i=m_i\\times x_i+c_i$, then for parallel lines,&nbsp;<strong>$m_1=m_2$</strong>.</p>\n<p>Vertical lines are always parallel to <strong>the $y$-axis</strong> and horizontal lines are always parallel to <strong>the $x$-axis</strong>.</p>\n<h4>b)</h4>\n<p>All but one of the straight lines above follow the general formula $y=mx+c$ where $m$ is a constant denoting the gradient of the line and $c$ is a constant denoting the $y$-intercept of the line.&nbsp;</p>\n<p>The gradient (given by $m$), indicates the slope of a straight line. Where this&nbsp;is positive, the $y$-coordinate along the line increases by $m$ for every increase in the value of $x$ by $1$. For example, $y=3x$ indicates that for every increase in the value of $x$ by $1$, the value of $y$ increases by $3$. Where there is no $m$ (i.e. $y=c$), the line is horizontal as the value of $y$ has no dependence on the value of $x$.</p>\n<p>The $y$-intercept (given by $c$), indicates the point where the line passes through the $y$-axis. If the line equation has no $c$ (i.e. $y=mx$), the line has a $y$ intercept of $0$ (passing through the origin).</p>\n<p>We can use our knowledge of the slope of the line and the $y$-intercept to then judge which points on the graph correspond to which equations.</p>\n<p><strong>A:</strong></p>\n<p>$y=\\simplify{{m1}x+{c1}}$ &nbsp;gives us $m=\\var{m1}$ and $c=\\var{c1}$.</p>\n<p><strong>B:</strong></p>\n<p>$y=\\simplify{{m2}x+{c2}}$ &nbsp;gives us $m=\\var{m2}$ and $c=\\var{c2}$.</p>\n<p><strong>C:</strong></p>\n<p>$y=\\simplify{{m3}x+{c3}}$ &nbsp;gives us $m=\\var{m3}$ and $c=\\var{c3}$.</p>\n<p><strong>D:</strong></p>\n<p>$y=\\simplify{{c4}}$ &nbsp;gives us $m=0$ and $c=\\var{c1}$.</p>\n<p><strong>E:</strong></p>\n<p>$x=\\var{c5}$ &nbsp;is the exception to the $y=mx+c$ form of writing straight line equations. This equation gives a value for $x$ independent of the value of $y$, so a vertical line is formed at the $x$-coordinate $\\var{c5}$.</p>\n<h4>c)</h4>\n<p>For the equation&nbsp;$y=\\var{cfake}$, the gradient is $0$, so we check our lines $A_0, B_0, C_0, D_0 \\text{ and }E_0$ to find $D_0$ has the same gradient. Therefore $D_0$ is parallel to $y=\\var{cfake}$.</p>\n<h4>d)</h4>\n<p>For the equation $y=\\var{m1}x+\\var{cpar}$, the gradient is $\\var{m1}$, so we check our lines $A_0, B_0, C_0, D_0 \\text{ and }E_0$ to find both $A_0$ and $B_0$ have the same gradient. Therefore $A_0$ and $B_0$ are&nbsp;parallel to $y=\\var{m1}x+\\var{cpar}$.</p>\n<h4>e)</h4>\n<p>This time, we have to find the gradient of the $x$-axis. Note that the $x$-axis is equivalent to the line with equation $y=0$. The gradient of this line is $0$, so we check our lines $A_0, B_0, C_0, D_0 \\text{ and }E_0$ to find $D_0$ has the same gradient. Therefore $D_0$ is parallel to $y=0$ and in other words, $D_0$ is parallel to the $x$-axis.</p>\n<h4>f)</h4>\n<p>For the equation $y=\\simplify{{muniq}x+{c1}}$, the gradient is $\\var{muniq}$, so we check our lines $A_0, B_0, C_0, D_0 \\text{ and }E_0$ to find none have the same gradient. Therefore none of the labelled lines are&nbsp;parallel to $y=\\simplify{{muniq}x+{c1}}$.</p>\n<h4>g)</h4>\n<p>Lines that are parallel have the same gradient. This is represented by $m$ in the general form of a straight line ($y=mx+c$). The gradient of $C_0$ was $\\var{m3}$, so any straight line parallel to $C_0$ will share this value for $m$.</p>\n<p>Therefore, we input $y=\\var{m3}x+c$.</p>", "statement": "<p>Use the graph bellow to wok out the line equations of lines $A_0 \\text{ to } E_0$ and use these equations to judge which lines they are parallel to.</p>\n<p>{geogebra_applet('WuavQqEG',[[\"m1\",m1],[\"m2\",m2],[\"m3\",m3],[\"c1\",c1],[\"c2\",c2],[\"c3\",c3],[\"c4\",c4],[\"c5\",c5],[\"f1\",f1],[\"d1\",d1],[\"a1\",a1],[\"b1\",b1]])}</p>", "variables": {"cpar": {"name": "cpar", "group": "intercept", "templateType": "anything", "description": "", "definition": "random(500..750)"}, "m1": {"name": "m1", "group": "Gradient", "templateType": "anything", "description": "", "definition": "random(1..2#0.1)"}, "c4": {"name": "c4", "group": "intercept", "templateType": "anything", "description": "", "definition": "random(3..6)"}, "c5": {"name": "c5", "group": "intercept", "templateType": "anything", "description": "", "definition": "random(14..17)"}, "cfake": {"name": "cfake", "group": "intercept", "templateType": "anything", "description": "", "definition": "random(-8..-3)"}, "m3": {"name": "m3", "group": "Gradient", "templateType": "anything", "description": "", "definition": "random(-2..-1#0.1)"}, "c3": {"name": "c3", "group": "intercept", "templateType": "anything", "description": "", "definition": "random(1..5)"}, "a1": {"name": "a1", "group": "Point variables(removed)", "templateType": "anything", "description": "", "definition": "30"}, "muniq": {"name": "muniq", "group": "Gradient", "templateType": "anything", "description": "", "definition": "random(10..50 except m1 m2 m3 mfake)"}, "c10": {"name": "c10", "group": "Point variables(removed)", "templateType": "anything", "description": "", "definition": "-30/m3"}, "m2": {"name": "m2", "group": "Gradient", "templateType": "anything", "description": "", "definition": "m1"}, "b1": {"name": "b1", "group": "Point variables(removed)", "templateType": "anything", "description": "", "definition": "25"}, "c2": {"name": "c2", "group": "intercept", "templateType": "anything", "description": "", "definition": "random(15..20)"}, "c1": {"name": "c1", "group": "intercept", "templateType": "anything", "description": "", "definition": "random(-10..-5)"}, "f1": {"name": "f1", "group": "Point variables(removed)", "templateType": "anything", "description": "", "definition": "-40"}, "mfake": {"name": "mfake", "group": "Gradient", "templateType": "anything", "description": "", "definition": "random(-4..-1)"}, "d1": {"name": "d1", "group": "Point variables(removed)", "templateType": "anything", "description": "", "definition": "60"}}, "tags": ["definition of parallel lines", "definitions", "equations of parallel lines", "parallel lines", "taxonomy"], "ungrouped_variables": [], "functions": {}, "name": "Equations of parallel lines", "preamble": {"js": "", "css": ""}, "extensions": ["geogebra"], "type": "question", "variable_groups": [{"variables": ["m1", "m2", "m3", "mfake", "muniq"], "name": "Gradient"}, {"variables": ["c1", "c2", "c3", "c4", "c5", "cfake", "cpar"], "name": "intercept"}, {"variables": ["a1", "b1", "c10", "d1", "f1"], "name": "Point variables(removed)"}], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Some conceptual questions about parallel lines - fill in the gaps in some statements.</p>\n<p>Identify the lines corresponding to given equations, and the lines parallel to lines with given equations.</p>\n<p></p>", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "matrix": ["1", 0, 0, 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", "", ""], "shuffleChoices": true, "marks": 0, "type": "1_n_2", "choices": ["<p>the same</p>", "<p>a variable</p>", "<p>a large</p>", "<p>a small</p>", "<p>a negligible</p>"], "showFeedbackIcon": true, "minMarks": 0}, {"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "matrix": ["1", 0, 0, 0, 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", "", "", ""], "shuffleChoices": true, "marks": 0, "type": "1_n_2", "choices": ["<p>gradient</p>", "<p>$y$-intercept</p>", "<p>length</p>", "<p>color</p>", "<p>$y$-coordinate</p>", "<p>$x$-coordinate</p>"], "showFeedbackIcon": true, "minMarks": 0}, {"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "matrix": ["1", 0, 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", ""], "shuffleChoices": false, "marks": 0, "type": "1_n_2", "choices": ["<p>m\u2081</p>", "<p>c\u2081</p>", "<p>x\u2081</p>", "<p>y\u2081</p>"], "showFeedbackIcon": true, "minMarks": 0}, {"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "matrix": ["1", 0, 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", ""], "shuffleChoices": false, "marks": 0, "type": "1_n_2", "choices": ["<p>m\u2082</p>", "<p>c\u2082</p>", "<p>x\u2082</p>", "<p>y\u2082</p>"], "showFeedbackIcon": true, "minMarks": 0}, {"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "matrix": ["0", "1", 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", ""], "shuffleChoices": false, "marks": 0, "type": "1_n_2", "choices": ["<p>the x-axis</p>", "<p>the y-axis</p>", "<p>y=x</p>", "<p>nothing</p>"], "showFeedbackIcon": true, "minMarks": 0}, {"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "matrix": ["1", 0, 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", ""], "shuffleChoices": false, "marks": 0, "type": "1_n_2", "choices": ["<p>the x-axis</p>", "<p>the y-axis</p>", "<p>y=x</p>", "<p>nothing</p>"], "showFeedbackIcon": true, "minMarks": 0}], "type": "gapfill", "prompt": "<p><em>Fill in the gaps below to complete the description of&nbsp;parallel lines.</em></p>\n<p>Parallel lines are&nbsp;defined as a lines that remain&nbsp;[[0]] distance apart and so they never meet. This means that their [[1]]&nbsp;is the same.</p>\n<p>If we write our line equations in the form $y=m_i\\times x_i+c_i$, then for parallel lines,&nbsp;[[2]] $=$&nbsp;[[3]].</p>\n<p>Vertical lines are always parallel to&nbsp;[[4]] and horizontal lines are always parallel to&nbsp;[[5]].</p>", "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"variableReplacements": [], "maxMarks": 0, "minMarks": 0, "minAnswers": 0, "answers": ["<p>$A_0$</p>", "<p>$B_0$</p>", "<p>$C_0$</p>", "<p>$D_0$</p>", "<p>$E_0$</p>", "<p>None of these</p>"], "displayType": "radiogroup", "warningType": "none", "layout": {"expression": "", "type": "all"}, "choices": ["<p>$y=\\simplify{{m1}x+{c1}}$</p>", "<p>$y=\\simplify{{m2}x+{c2}}$</p>", "<p>$y=\\simplify{{m3}x+{c3}}$</p>", "<p>$y=\\var{c4}$</p>", "<p>$x=\\var{c5}$</p>", "<p>$y=\\simplify{{mfake}x+{cfake}}$</p>"], "scripts": {}, "showCorrectAnswer": true, "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": true, "marks": 0, "prompt": "<p>Use the graph above to match the correct line to each equation.</p>", "matrix": [["1", 0, 0, 0, 0, 0], [0, "1", 0, 0, 0, 0], [0, 0, "1", 0, 0, 0], [0, 0, 0, "1", 0, 0], [0, 0, 0, 0, "1", 0], [0, 0, 0, 0, 0, "1"]], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "shuffleAnswers": false}, {"variableReplacements": [], "maxMarks": 0, "minMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "checkbox", "warningType": "none", "matrix": [0, 0, 0, "1", 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", "", "", ""], "maxAnswers": 0, "shuffleChoices": false, "marks": 0, "prompt": "<p>Which lines are parallel to the line with equation $y=\\var{cfake}$?</p>", "type": "m_n_2", "choices": ["<p>$A_0$</p>", "<p>$B_0$</p>", "<p>$C_0$</p>", "<p>$D_0$</p>", "<p>$E_0$</p>", "<p>None of these</p>"], "showFeedbackIcon": true, "minAnswers": 0}, {"variableReplacements": [], "maxMarks": "1", "minMarks": "0", "variableReplacementStrategy": "originalfirst", "displayType": "checkbox", "warningType": "none", "matrix": ["1/2", "1/2", 0, "0", 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", "", "", ""], "maxAnswers": 0, "shuffleChoices": false, "marks": 0, "prompt": "<p>Which lines&nbsp;are parallel to the line with equation $y=\\var{m1}x+\\var{cpar}$?</p>", "type": "m_n_2", "choices": ["<p>$A_0$</p>", "<p>$B_0$</p>", "<p>$C_0$</p>", "<p>$D_0$</p>", "<p>$E_0$</p>", "<p>None of these</p>"], "showFeedbackIcon": true, "minAnswers": 0}, {"variableReplacements": [], "maxMarks": 0, "minMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "checkbox", "warningType": "none", "matrix": ["0", "0", 0, "2", "0", 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", "", "", ""], "maxAnswers": 0, "shuffleChoices": false, "marks": 0, "prompt": "<p>Which lines&nbsp;are parallel to the $x$ axis?</p>", "type": "m_n_2", "choices": ["<p>$A_0$</p>", "<p>$B_0$</p>", "<p>$C_0$</p>", "<p>$D_0$</p>", "<p>$E_0$</p>", "<p>None of these</p>"], "showFeedbackIcon": true, "minAnswers": 0}, {"variableReplacements": [], "maxMarks": 0, "minMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "checkbox", "warningType": "none", "matrix": ["0", "0", 0, "0", "0", "1"], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", "", "", ""], "maxAnswers": 0, "shuffleChoices": false, "marks": 0, "prompt": "<p>Which lines&nbsp;are parallel to the&nbsp;line with the equation $y=\\simplify{{muniq}x+{c1}}$?</p>", "type": "m_n_2", "choices": ["<p>$A_0$</p>", "<p>$B_0$</p>", "<p>$C_0$</p>", "<p>$D_0$</p>", "<p>$E_0$</p>", "<p>None of these</p>"], "showFeedbackIcon": true, "minAnswers": 0}, {"scripts": {}, "showCorrectAnswer": true, "gaps": [{"vsetrange": [0, 1], "scripts": {}, "variableReplacements": [], "expectedvariablenames": [], "answer": "{m3}", "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "showCorrectAnswer": true, "type": "jme", "marks": 1, "showpreview": true, "vsetrangepoints": 5, "checkvariablenames": false, "checkingaccuracy": 0.001, "showFeedbackIcon": true}], "type": "gapfill", "prompt": "<p>Complete&nbsp;the general equation in the form $y=mx+c$ for any line parallel to line $C_0$.</p>\n<p>$y=$[[0]]$x+c$</p>", "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}]}