// Numbas version: exam_results_page_options
{"name": "Basic algebra practice", "metadata": {"description": "<p>Extra practice on some basic algebra skills, including solving linear equations. You can try as many times as you like and also generate new versions of the questions for extra practice.</p>", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Simplify: Combine Like Terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Simple exercise in collecting terms in different powers of \\(x\\)</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h3>Simplify the following expression by combining \"like\" terms.</h3>", "advice": "<p>The idea is to collect together and combine any terms that are the same kind of term so:</p>\n<p>$\\var{b}$ and $\\var{f}$ are ordinary numbers. We can combine them to get $\\var{b+f}$</p>\n<p>We can combine $\\var{a}x$ and $\\var{d}x$ to get $\\var{a+d}x$.</p>\n<p>There are also $\\var{c}$ times $x^2$. So our answer is:</p>\n<p>$\\simplify{{c}x^2+{a+d}x+{b+f}}$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(-5..10 except 0 except -b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "f"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify[!collectNumbers]{{a}x+{b}+{c}x^2+{d}x+{f}}$</p>", "answer": "{c}x^2+({a}+{d})x+({b}+{f})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`+-$n`?*x^2+`+-$n`?*x+`+-$n`?", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Equations: Linear equations 1a - rational answers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Solve linear equations with unknowns on one. Including brackets and fractions.</p>", "licence": "None specified"}, "statement": "<p>Solve the following equations to find $x$:</p>", "advice": "<h3><strong>part a)</strong></h3>\n<p>\\[\\simplify{{a}x+{b} = {c}} \\]</p>\n<p>The first step is to isolate all the $x$-terms to one side of the equation.</p>\n<p>To do that we will {add} $\\var{abs(b)}$ onto both sides:</p>\n<p>\\[\\var{a}x &nbsp;= \\simplify{{c-b}} \\]</p>\n<p>Now we need to divide both sides by the coefficient of $x$, to leave just one $x$.</p>\n<p>Dividing both sides by $\\var{a}$,</p>\n<p>\\[ &nbsp;x = {\\simplify{({c}-{b})/{a}}} \\]</p>\n<p></p>\n<h3><strong>part b)</strong></h3>\n<p>&nbsp;\\[ \\frac{\\simplify{{d}x + {f}}}{\\var{g}} = \\var{h} \\]<br/><br/>The first step is to rearrange by removing the fraction on the left. &nbsp;To do this we mulitply both sides by $\\var{g}$.<br/><br/>\\[ \\begin{split} \\frac{\\simplify{{d}x + {f}}}{\\var{g}} \\times \\var{g}&nbsp; &amp;= \\var{h} \\times \\var{g} \\\\\\\\ \\simplify{{d}x + {f}} &amp;= \\var{h*g} \\end{split} \\]<br/><br/>Next we&nbsp;isolate all the $x$-terms onto one side of the equation.<br/><br/>To do that we will {add2} $\\var{abs(f)}$ on both sides:<br/><br/>\\[ \\begin{split}&nbsp;\\var{d}x &amp;= \\simplify[]{{h*g}-{f}} \\\\ &amp;= \\simplify[]{{h*g-f}} \\end{split}\\]<br/><br/>Finally we need to divide both sides by the coefficient of $x$, to leave just one $x$.</p>\n<p>Dividing both sides by $\\var{d}$,<br/><br/>\\[x = \\simplify[fractionNumbers]{{(h*g-f)/d}} &nbsp;\\]</p>\n<p></p>\n<h3>part c)</h3>\n<p>\\[ \\simplify{{b}({c}x+{g})} = \\var{d} \\]</p>\n<p>Even though this looks different, this is quite similar to part b). &nbsp;We just have a multiplication rather than a division to deal with as the first step.<br/><br/>Rearrange to remove the multiplication on the left. &nbsp;To do this we divide both sides by $\\var{b}$:<br/><br/>\\[ \\begin{split} \\frac{\\simplify{{b}({c}x+{g})}}{ \\var{b}} &amp;= \\frac{\\var{d}}{\\var{b}} \\\\ \\\\ \\simplify{{c}x+{g}} &amp;= \\simplify[fractionNumbers]{{d/b}} \\end{split} \\]<br/><br/><br/>Next we&nbsp;isolate all the $x$-terms onto one side of the equation.<br/><br/>To do that we will {add3} $\\var{abs(g)}$ on both sides:<br/><br/>\\[ \\begin {split} \\simplify{{c}x + {g}} &nbsp;&nbsp;\\var{add3sym} &nbsp; \\var{abs(g)} &amp;= \\simplify[fractionNumbers]{{d/b}}&nbsp;\\var{add3sym} \\var{abs(g)} \\\\ \\var{c}x &amp;= \\simplify[fractionNumbers]{{d/b-g}} \\end{split} \\]<br/><br/>Finally we need to divide both sides by the coefficient of $x$, to leave just one $x$</p>\n<p>Dividing both sides by $\\var{c}$:<br/><br/>\\[ x = &nbsp;\\simplify[fractionNumbers]{{(d/b-g)/c}} \\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-20..20 except 0 except 1 except -1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-20..20 except [0,1,-1])", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-20..20 except [-1,0,1])", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-20..20 except 0 except 1 except -1)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(-20..20 except 0)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "random(-20..20 except 0 except 1 except -1)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(-20..20 except 0)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-20..20 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-20..20 except 0)", "description": "", "templateType": "anything", "can_override": false}, "add": {"name": "add", "group": "Ungrouped variables", "definition": "if(b>0, 'subtract', 'add')", "description": "", "templateType": "anything", "can_override": false}, "addsym": {"name": "addsym", "group": "Ungrouped variables", "definition": "if(add='add',' + ',' - ')", "description": "", "templateType": "anything", "can_override": false}, "add2": {"name": "add2", "group": "Ungrouped variables", "definition": "if(f>0, 'subtract', 'add')", "description": "", "templateType": "anything", "can_override": false}, "add2sym": {"name": "add2sym", "group": "Ungrouped variables", "definition": "if(add2='add',' + ',' - ')", "description": "", "templateType": "anything", "can_override": false}, "add3": {"name": "add3", "group": "Ungrouped variables", "definition": "if(g>0, 'subtract', 'add')", "description": "", "templateType": "anything", "can_override": false}, "add3sym": {"name": "add3sym", "group": "Ungrouped variables", "definition": "if(add3='add',' + ',' - ')", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "((c-b)/a)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "((h*g - f)/d)", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "(((d/b) - g)/c)", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "(f-c)/(h-m)", "description": "", "templateType": "anything", "can_override": false}, "a5": {"name": "a5", "group": "Ungrouped variables", "definition": "((g*m-n*f)/(n*b-g*c))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "h", "m", "n", "add", "addsym", "add2", "add2sym", "add3", "add3sym", "a1", "a2", "a3", "a4", "a5"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "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>$\\simplify{{a}x+{b} = {c}}$</p>\n<p></p>\n<p>$x=$ [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "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>$\\dfrac{\\simplify{{d}x + {f}}}{\\var{g}} = \\var{h}$</p>\n<p></p>\n<p>$x=$ [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "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>$\\simplify{{b}({c}x+{g})} = \\var{d}$</p>\n<p></p>\n<p>$x=$ [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a3}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Equations: Linear equations 2 - unknowns on both sides", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "heike hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2960/"}, {"name": "sean hunte", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3167/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Solve linear equations with unkowns on both sides. Including brackets and fractions.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "<p><strong>Part a)</strong></p>\n<p>Given $\\simplify{{l}({m}w-{n}) = {p}w+{q}}$, we can expand the brackets, get all the $w$'s on the left hand side and all the numbers on the right hand side, and then divide both sides by the coefficient of $w$ to get $w$ by itself.</p>\n<p></p>\n<table style=\"height: 342px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\">$\\simplify{{l}({m}w-{n})}$</td>\n<td style=\"height: 18px; width: 29.2656px;\">$=$</td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\">$\\simplify{{p}w+{q}}$&nbsp;</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\">&nbsp;</td>\n<td style=\"height: 18px; width: 29.2656px;\"></td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\"></td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\">$\\simplify{{l*m}w+{n*l}}$</td>\n<td style=\"height: 18px; width: 29.2656px;\">$=$</td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\">$\\simplify{{p}w+{q}}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\">&nbsp;</td>\n<td style=\"height: 18px; width: 29.2656px;\"></td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\"></td>\n</tr>\n<tr style=\"height: 36px;\">\n<td style=\"text-align: right; height: 36px; width: 218.312px;\">$\\simplify[!cancelTerms,unitFactor]{{l*m}w-{n*l}-{p}w}$</td>\n<td style=\"height: 36px; width: 29.2656px;\">$=$</td>\n<td style=\"text-align: left; height: 36px; width: 283.422px;\">$\\simplify[!cancelTerms,unitFactor]{{p}w+{q}-{p}w}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\"></td>\n<td style=\"height: 18px; width: 29.2656px;\"></td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\">&nbsp;</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\">$\\simplify{{l*m-p}w-{n*l}}$</td>\n<td style=\"height: 18px; width: 29.2656px;\">$=$</td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\">$\\var{q}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\"></td>\n<td style=\"height: 18px; width: 29.2656px;\"></td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\">&nbsp;</td>\n</tr>\n<tr style=\"height: 36px;\">\n<td style=\"text-align: right; height: 36px; width: 218.312px;\">$\\var{l*m-p}w-\\var{n*l}+\\var{n*l}$</td>\n<td style=\"height: 36px; width: 29.2656px;\">$=$</td>\n<td style=\"text-align: left; height: 36px; width: 283.422px;\">$\\var{q}+\\var{n*l}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\"></td>\n<td style=\"height: 18px; width: 29.2656px;\"></td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\">&nbsp;</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\">$\\var{l*m-p}w$</td>\n<td style=\"height: 18px; width: 29.2656px;\">$=$</td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\">$\\var{q+n*l}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\"></td>\n<td style=\"height: 18px; width: 29.2656px;\"></td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\">&nbsp;</td>\n</tr>\n<tr style=\"height: 36px;\">\n<td style=\"text-align: right; height: 36px; width: 218.312px;\">$\\displaystyle{\\frac{\\var{l*m-p}w}{\\var{l*m-p}}}$</td>\n<td style=\"height: 36px; width: 29.2656px;\">$=$</td>\n<td style=\"text-align: left; height: 36px; width: 283.422px;\">$\\displaystyle{\\frac{\\var{q+n*l}}{\\var{l*m-p}}}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"text-align: right; height: 18px; width: 218.312px;\"></td>\n<td style=\"height: 18px; width: 29.2656px;\"></td>\n<td style=\"text-align: left; height: 18px; width: 283.422px;\">&nbsp;</td>\n</tr>\n<tr style=\"height: 36px;\">\n<td style=\"text-align: right; height: 36px; width: 218.312px;\">$w$</td>\n<td style=\"height: 36px; width: 29.2656px;\">$=$</td>\n<td style=\"text-align: left; height: 36px; width: 283.422px;\">$\\displaystyle{\\simplify{{q+n*l}/{l*m-p}}} = \\var{precround(ansA,1)} \\text{ to 1 dp}$</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>Part b)</strong></p>\n<p>Given $\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}=\\var{g}$, we can multiply both sides by $(y-\\var{f})$ to get rid of the fraction, get all the $y$'s on one side and the numbers on the other side, and then divide both sides by the coefficient of $y$&nbsp;to get $y$ by itself.</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\var{g}$&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}\\times(y-\\var{f})$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\var{g}\\times (y-\\var{f})$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\var{d}y$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[unitFactor]{{g}y+{-g*f}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify[!cancelTerms,unitFactor]{{d}y+{-g}y}$&nbsp;</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[!cancelTerms,unitFactor]{{g}y+{-g*f}+{-g}y}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\var{d-g}y$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\var{-g*f}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\frac{\\var{d-g}y}{\\var{d-g}}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\frac{\\var{-g*f}}{\\var{d-g}}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;$y$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle{\\simplify{{-g*f}/{d-g}}}= \\var{precround(ansB,1)}\\text{ to 1 dp}$</td>\n</tr>\n</tbody>\n</table>\n<p><strong>part c)</strong></p>\n<p>Given $\\displaystyle{\\frac{x+\\var{add}}{\\var{denom1}}+\\frac{x}{\\var{denom2}}=\\var{right}}$, we can multiply both sides by $\\var{denom1}$ and by&nbsp;$\\var{denom2}$ to get rid of the fractions, get all the $x$'s on one side and the numbers on the other side, and then divide both sides by the coefficient of $x$&nbsp;to get $x$ by itself.</p>\n<p></p>\n<table style=\"height: 403px;\">\n<tbody>\n<tr style=\"height: 34px;\">\n<td style=\"text-align: right; height: 34px; width: 565.122314453125px;\">$\\displaystyle{\\frac{x+\\var{add}}{\\var{denom1}}+\\frac{x}{\\var{denom2}}}$</td>\n<td style=\"height: 34px; width: 29.076087951660156px;\">$=$</td>\n<td style=\"text-align: left; height: 34px; width: 341.98370361328125px;\">$\\var{right}$&nbsp;</td>\n<td style=\"text-align: left; height: 34px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\">&nbsp;</td>\n<td style=\"height: 17px; width: 29.076087951660156px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 52px;\">\n<td style=\"text-align: right; height: 52px; width: 565.122314453125px;\">$\\displaystyle{\\left(\\frac{x+\\var{add}}{\\var{denom1}}\\right)\\times\\var{denom1}+\\left(\\frac{x}{\\var{denom2}}\\right)\\times\\var{denom1}}$</td>\n<td style=\"height: 52px; width: 29.076087951660156px;\">$=$</td>\n<td style=\"text-align: left; height: 52px; width: 341.98370361328125px;\">$\\var{right}\\times \\var{denom1}$</td>\n<td style=\"text-align: left; height: 52px; width: 126.83424377441406px;\">(multiply all terms by $\\var{denom1}$)</td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\">&nbsp;</td>\n<td style=\"height: 17px; width: 29.076087951660156px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\">$\\displaystyle{x+\\var{add}+\\frac{\\var{denom1}x}{\\var{denom2}}}$</td>\n<td style=\"height: 17px; width: 29.076087951660156px;\">$=$</td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\">$\\var{r1}$</td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\"></td>\n<td style=\"height: 17px; width: 29.076087951660156px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\">&nbsp;</td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 52px;\">\n<td style=\"text-align: right; height: 52px; width: 565.122314453125px;\">$\\displaystyle{(x+\\var{add})\\times\\var{denom2}+\\left(\\frac{\\var{denom1}x}{\\var{denom2}}\\right)\\times\\var{denom2}}$&nbsp;</td>\n<td style=\"height: 52px; width: 29.076087951660156px;\">$=$</td>\n<td style=\"text-align: left; height: 52px; width: 341.98370361328125px;\">$\\var{r1}\\times\\var{denom2}$</td>\n<td style=\"text-align: left; height: 52px; width: 126.83424377441406px;\">(multiply all terms by $\\var{denom2}$)</td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\"></td>\n<td style=\"height: 17px; width: 29.076087951660156px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\">&nbsp;</td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\">$\\displaystyle{\\var{denom2}x+\\var{a2}+\\var{denom1}x}$</td>\n<td style=\"height: 17px; width: 29.076087951660156px;\">$=$</td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\">$\\var{r12}$</td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\"></td>\n<td style=\"height: 17px; width: 29.076087951660156px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\">&nbsp;</td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 34px;\">\n<td style=\"text-align: right; height: 34px; width: 565.122314453125px;\">$\\var{sumdeno}x+\\var{a2}$</td>\n<td style=\"height: 34px; width: 29.076087951660156px;\">$=$</td>\n<td style=\"text-align: left; height: 34px; width: 341.98370361328125px;\">$\\var{r12}$</td>\n<td style=\"text-align: left; height: 34px; width: 126.83424377441406px;\">(collect like terms)</td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\"></td>\n<td style=\"height: 17px; width: 29.076087951660156px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\">&nbsp;</td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 34px;\">\n<td style=\"text-align: right; height: 34px; width: 565.122314453125px;\">&nbsp;$\\var{sumdeno}x$</td>\n<td style=\"height: 34px; width: 29.076087951660156px;\">$=$</td>\n<td style=\"text-align: left; height: 34px; width: 341.98370361328125px;\">$\\var{r12}-\\var{a2}$</td>\n<td style=\"text-align: left; height: 34px; width: 126.83424377441406px;\">(collect like terms)</td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\"></td>\n<td style=\"height: 17px; width: 29.076087951660156px;\">&nbsp;</td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\">$\\var{sumdeno}x$</td>\n<td style=\"height: 17px; width: 29.076087951660156px;\">$=$</td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\">$\\var{top}$</td>\n<td style=\"text-align: left; height: 17px; width: 126.83424377441406px;\"></td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"text-align: right; height: 17px; width: 565.122314453125px;\"></td>\n<td style=\"height: 17px; width: 29.076087951660156px;\"></td>\n<td style=\"text-align: left; height: 17px; width: 341.98370361328125px;\">&nbsp;</td>\n<td style=\"text-align: left; height: 17px; width: 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