// Numbas version: finer_feedback_settings
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interpreted_answers\n \ngap_feedback (Feedback on each of the gaps):\n    map(\n        try(\n            let(\n                answer, studentAnswer[answer_number],\n                result, submit_part(gaps[gap_number][\"path\"],answer),\n                gap, gaps[gap_number],\n                name, gap[\"name\"],\n                noFeedbackIcon, not gap[\"settings\"][\"showFeedbackIcon\"],\n                non_warning_feedback, filter(x[\"op\"]<>\"warning\",x,result[\"feedback\"]),\n                    assert(noFeedbackIcon,\n                        assert(name=\"\" or len(gaps)=1 or len(non_warning_feedback)=0,feedback(translate('part.gapfill.feedback header',[\"name\": name])))\n                    );\n                    concat_feedback(non_warning_feedback, if(marks>0,result[\"marks\"]/marks,1/len(gaps)), noFeedbackIcon);\n                    result\n            ),\n            err,\n            fail(translate(\"part.gapfill.error marking gap\",[\"name\": gaps[gap_number][\"name\"], \"message\": err]))\n        ),\n        [gap_number, answer_number],\n        zip(gap_adaptive_order, gap_adaptive_order)\n    )\n \nall_valid (Are the answers to all of the gaps valid?):\n  all(map(res[\"valid\"], res, marked_original_order))\n \ncheck_answers:\nif(\n   answers[2]=answers[0],\n   sub_credit(1/2, \"Numbers cannot be the same. \"),\n   feedback(\"\")\n)\n \nmark:\n  assert(all_valid or not settings[\"sortAnswers\"], fail(translate(\"question.can not submit\")));\n  apply(answers);\n  apply(check_answers);\n  apply(gap_feedback)\n \ninterpreted_answer:\n  answers\n \npre_submit:\n    map(\n        let(\n            answer, studentAnswer[answer_number],\n            result, submit_part(gaps[gap_number][\"path\"],answer),\n            check_pre_submit(gaps[gap_number][\"path\"], answer, exec_path)\n        ),\n        [gap_number,answer_number],\n        zip(gap_order,answer_order)\n    )", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the $x$ and $y$ values that satisfy both of the following equations. That is, find the points of intersection of the two curves.</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{grad}x+{yint}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(1)$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{num}/x+{d}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(2)$</td>\n</tr>\n</tbody>\n</table>\n<p>$x_1=$&nbsp;[[0]], &nbsp; $y_1=$&nbsp;[[1]] and $x_2=$ [[2]], $y_2=$&nbsp;[[3]]</p>\n<p></p>\n<p></p>", "stepsPenalty": "4", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Given</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{grad}x+{yint}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(1)$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{num}/x+{d}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$(2)$</td>\n</tr>\n</tbody>\n</table>\n<p>substitute the expression for $y$ given in $(1)$ into $(2)$:<br/>\\[\\simplify{{grad}x+{yint} ={num}/x+{d}}\\]</p>\n<p>To get rid of the&nbsp;$x$ in the denominator, let us multiply both sides by $x$<br/>\\[\\simplify{{grad}x^2+{yint}x ={num}+{d}x}\\]</p>\n<p>Notice this equation is a quadratic, we put everything on one side</p>\n<p>\\[\\simplify{{grad}x^2+{yint-d}x -{num}=0}\\]</p>\n<p>There are various ways to solve a quadratic, in this particular case we can factorise the quadratic:</p>\n<p>\\[(\\simplify{{a}x+{b}})(\\simplify{x+{c}})=0\\]</p>\n<p>Therefore,&nbsp;$x=\\simplify{{-b}/{a}},\\,\\var{-c}$.</p>\n<p><br/>Now for&nbsp;$x=\\simplify[fractionnumbers]{{root1}}$, we can determine the corresponding $y$ value by substituting $x=\\simplify[fractionnumbers]{{root1}}$ into either equation $(1)$ or $(2)$. Below we substitute into $(1)$:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify[!collectnumbers,fractionnumbers]{{grad}({root1})+{yint}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{ansy1}$</td>\n</tr>\n</tbody>\n</table>\n<p>Now for&nbsp;$x=\\simplify[fractionnumbers]{{root2}}$, so we can determine the corresponding $y$ value by substituting $x=\\simplify[fractionnumbers]{{root2}}$ into either equation $(1)$ or $(2)$.&nbsp;Below we substitute into $(1)$:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify[!collectnumbers,fractionnumbers]{{grad}({root2})+{yint}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{ansy2}$</td>\n</tr>\n</tbody>\n</table>\n<p>Therefore the values that satisfy equations $(1)$ and $(2)$ are $x_1=\\simplify[fractionnumbers]{{root1}}$,&nbsp;$y_1=\\var{ansy1}$ and&nbsp;$x_2=\\simplify[fractionnumbers]{{root1}}$,&nbsp;$y_2=\\var{ansy2}$.</p>\n<p>In other words, the two curves intersect at the points $\\left(\\simplify[fractionnumbers]{{root1}},\\var{ansy1}\\right)$ and&nbsp;$\\left(\\simplify[fractionnumbers]{{root2}},\\var{ansy2}\\right)$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": false, "customName": "", 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