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Perpendicular gradients have product $-1$, so take the **negative reciprocal**: turn the fraction upside down and change its sign.\n\nTherefore the perpendicular gradient is $m=\\frac{2}{3}$. Identifying this earns **M1**.","highlights":[{"text":"perpendicular to $y = -\\frac{3}{2}x - 1$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@orange}{\\htmlData{anchor=y_value}{y}}=\\frac{2}{3}\\color{@green}{\\htmlData{anchor=x_value}{x}}+c$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=minus_eight}{-8}}=\\frac{2}{3}\\color{@green}{\\htmlData{anchor=zero}{(0)}}+c$"},{"latex":"$$\\color{@purple}{\\htmlData{anchor=intercept}{c=-8}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"y_value"},{"color":"orange","style":"text-color","anchor":"minus_eight"},{"color":"green","style":"text-color","anchor":"x_value"},{"color":"green","style":"text-color","anchor":"zero"}]},"order":1,"title":"Use the given point","description":"Use the gradient-intercept form. At the point (0, -8), substitute x = 0 and y = -8 to find the intercept."},{"type":"text","order":2,"title":"State the equation","content":"Insert $m=\\frac{2}{3}$ and $c=-8$ into $y=mx+c$:\n\n$$\\boxed{y=\\frac{2}{3}x-8}$$\n\nThis correct equation earns **A1**. Notice that the $-1$ from the original line is not used: the required line has its own intercept because it passes through $(0,-8)$.","highlights":[{"text":"passes through $(0,-8)$","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:38:02.409Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":94},"walkthroughVideo":null},{"id":"e0cb84a5-5545-4566-9b6a-7e816ee630e0","revisionId":"51d16718-8b22-4efb-8f53-bcac7c6426ab","questionNumber":"7","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Parallel and Perpendicular Lines","calculatorUse":null,"isCalculation":true,"parts":[{"id":"8c5b26c1-a6d6-4463-b49f-4e5007bc8c75","label":"0","content":"Find the equation of the line parallel to $2y - 3x + 2 = 0$ which passes through $(0,4)$","markscheme":"- Rearranges to find gradient: $2y = 3x - 2 \\implies y = \\frac{3}{2}x - 1$, so $m = \\frac{3}{2}$ M1\n- Correct equation $y = \\frac{3}{2}x + 4$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Rearranges the given equation to find the gradient $m = \\frac{3}{2}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct final equation $y = \\frac{3}{2}x + 4$.","code":"A"}],"finalAnswers":["y=3/2x+4"],"generatedAt":"2026-05-07T02:03:01.197Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"gradient_intercept","label":"Use gradient and intercept","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=original}{2y-3x+2=0}$"},{"latex":"$$\\htmlData{anchor=rearranged}{2y=3x-2}$"},{"latex":"$$\\htmlData{anchor=standard_form}{y=\\color{@sky}{\\htmlData{anchor=gradient}{\\frac{3}{2}}}x-1}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"gradient"}],"connections":[{"to":"rearranged","from":"original","color":"sky","label":"rearrange"},{"to":"standard_form","from":"rearranged","color":"orange","label":"divide by 2"}]},"order":1,"title":"Rearrange the line","description":"Rearranging into y = mx + c gives the gradient 3/2 (M1)."},{"type":"text","order":2,"title":"Use the same gradient","content":"Parallel lines have the same gradient, so the new line has the form $y=\\frac{3}{2}x+c$.","highlights":[{"text":"parallel to $2y - 3x + 2 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=form}{y=\\frac{3}{2}x+c}$"},{"latex":"$$\\htmlData{anchor=substitution}{4=\\frac{3}{2}(0)+c}$"},{"latex":"$$\\htmlData{anchor=intercept}{c=4}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"intercept"}],"connections":[{"to":"substitution","from":"form","color":"amber","label":"substitute the point"},{"to":"intercept","from":"substitution","color":"sky","label":"simplify"}]},"order":3,"title":"Substitute the point","description":"Use the point with coordinates (0, 4) to find the intercept."},{"type":"text","order":4,"title":"State the equation","content":"The required equation is $y=\\frac{3}{2}x+4$ (A1). Check: when $x=0$, this gives $y=4$, so it passes through $(0,4)$.","highlights":[{"text":"which passes through $(0,4)$","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:55:43.561Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":84},"walkthroughVideo":null},{"id":"91efa1cb-1437-4764-928f-184d64d102eb","revisionId":"689dbc64-9954-4572-b6fa-be8e7e1f0616","questionNumber":"9","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Parallel and Perpendicular Lines","calculatorUse":null,"isCalculation":true,"parts":[{"id":"1d9470b5-a4eb-4d3a-ae49-ff97d3974555","label":"0","content":"Find the equation of the line perpendicular to $5y = 2x - 4$ which passes through $(0,7)$","markscheme":"- Finds original gradient $m = \\frac{2}{5}$ and perpendicular gradient $m = -\\frac{5}{2}$ M1\n- Correct equation $y = -\\frac{5}{2}x + 7$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Finds the perpendicular gradient $m = -\\frac{5}{2}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct final equation $y = -\\frac{5}{2}x + 7$.","code":"A"}],"finalAnswers":["y=-5/2x+7"],"generatedAt":"2026-05-07T02:03:01.197Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":92},"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","summaries":[{"id":"acbe8ebb-46d9-478e-8f4d-932b2885f04c","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":true},{"id":"00df6036-9aab-4beb-a1f9-478482a7e43b","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"88754053-ce58-485c-a539-99de2a7000a2","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"99e9a58e-58ce-4c5b-bcf8-e1fc35241405","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":true},{"id":"5aa9fb76-4e85-46d4-9eb8-828c02ccc7a6","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"9b5feb40-0aa0-44e4-8ce6-453faebd020a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e0cb84a5-5545-4566-9b6a-7e816ee630e0","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"d258283c-865e-47c0-89da-0e5a1bcf9a81","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"91efa1cb-1437-4764-928f-184d64d102eb","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"6277a69d-6c9b-4e4d-a024-c4c43440a622","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"e3704fe4-d3b3-4d05-a5be-25f12b806bf9","difficultyBand":"medium","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"3e371dc2-7407-45d1-80db-f83b8384f7ae","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"279118c0-d7d5-42b6-8a6a-0acb5f408d8f","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c5bf0b85-9272-4d46-9a99-b4887b1793ec","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4bb0fdaa-3145-4c69-b31a-7058b11ddbd8","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a8a661f3-569f-4c77-a493-be9716343848","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c22eabf2-eada-4b44-bdb5-ceda01fdd864","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f5aac4ae-4ceb-4ac8-9cdc-ca59354e843c","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2db8e411-8c21-448e-ae58-f85a6e6f3962","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true}],"title":"Parallel and Perpendicular Lines","topicId":"d54ee42b-2549-46dd-a4b8-64a2f54ad90f","topicSidebar":"$L3c","topicSlug":"grade-6-6-12-parallel-and-perpendicular-lines","topicTitle":"Parallel and Perpendicular Lines"}]}],"$L3d","$L3e"]}]