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This is called *elimination*: combining equations to remove one variable.","highlights":[{"text":"$$$3x - 2y = 7$$","location":"specification"},{"text":"$$$7x + 2y = 13$$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=add}{\\color{@sky}{(3x-2y)+(7x+2y)=7+13}}"},{"latex":"\\htmlData{anchor=result}{\\color{@green}{10x=20}}"},{"text":"M1: adding gives 10x = 20."}],"highlights":[{"color":"sky","style":"text-color","anchor":"add"},{"color":"green","style":"text-color-with-background","anchor":"result"}],"connections":[{"to":"result","from":"add","color":"sky","label":"add and simplify","style":"solid"}]},"order":1,"title":"Add the equations","description":"Add the left-hand sides and add the right-hand sides. The y terms cancel, leaving an equation in x only."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=start}{\\color{@sky}{10x}=20}"},{"latex":"\\htmlData{anchor=answer}{\\color{@green}{x=2}}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"start"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"start","color":"sky","label":"divide by 10","style":"solid"}]},"order":2,"title":"Find x","description":"Divide both sides by 10 to isolate x."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=substitute}{3(2)-2y=7}"},{"latex":"\\htmlData{anchor=rearrange}{-2y=1}"},{"latex":"\\htmlData{anchor=answer}{\\color{@green}{y=-\\frac{1}{2}}}"},{"text":"M1: finding x = 2 and then substituting it to find y earns the second method mark. A1: x = 2 and y = -1/2."}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"rearrange","from":"substitute","color":"sky","label":"subtract 6","style":"solid"},{"to":"answer","from":"rearrange","color":"sky","label":"divide by -2","style":"solid"}]},"order":3,"title":"Substitute for y","description":"Put the value 2 for x into the first original equation, then rearrange to find y."}]}],"generatedAt":"2026-07-21T04:07:08.362Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"26393c00-9646-48ae-9f1f-7cedf8bec494","revisionId":"26655113-d5a6-4a0f-8099-daa2f97ad7fd","questionNumber":"7","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Simultaneous Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"30c6adf3-0371-491c-b185-95b3c8989fcd","label":"0","content":"Solve the simultaneous equations\n\n$$2x - 3y = 4$$\n\n$$4x - y = 13$$","markscheme":"- Multiplies first equation by 2: $4x - 6y = 8$ M1\n- Subtracts equations to find $y = 1$ M1\n- Substitutes $y = 1$ to find $x = \\frac{7}{2}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Multiplies first equation by 2 or second by 3 to equate coefficients.","code":"M"},{"id":"p2","marks":1,"description":"Eliminates a variable and finds $y = 1$ or $x = 3.5$.","code":"M"},{"id":"p3","marks":1,"description":"Correct values: $x = 3.5$ and $y = 1$.","code":"A"}],"finalAnswers":["x=3.5, y=1"],"generatedAt":"2026-05-07T01:58:24.365Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"elimination","label":"Elimination","steps":[{"type":"text","order":0,"title":"Match the x terms","content":"Use **elimination**: make one variable have matching coefficients, then subtract an equation to remove it. Here, multiplying the first equation by $2$ makes its $x$-coefficient match the $4x$ in the second equation.","highlights":[{"text":"Solve the simultaneous equations","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{\\color{@sky}{2x-3y=4}}"},{"latex":"\\htmlData{anchor=scaled}{\\color{@orange}{4x-6y=8}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"original"},{"color":"orange","style":"text-color-with-background","anchor":"scaled"}],"connections":[{"to":"scaled","from":"original","color":"orange","label":"multiply by 2"}]},"order":1,"title":"Scale the first equation","description":"Multiply every term in the first equation by 2, including the right-hand side. This earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=subtraction}{\\color{@sky}{(4x-6y=8)-(4x-y=13)}}"},{"latex":"\\htmlData{anchor=yequation}{\\color{@orange}{-5y=-5}}"},{"latex":"\\htmlData{anchor=yvalue}{\\color{@green}{y=1}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"subtraction"},{"color":"green","style":"text-color-with-background","anchor":"yvalue"}],"connections":[{"to":"yequation","from":"subtraction","color":"orange","label":"subtract"},{"to":"yvalue","from":"yequation","color":"green","label":"divide by -5"}]},"order":3,"title":"Eliminate x","description":"Subtract the second equation from the scaled first equation. The 4x terms cancel, and finding y = 1 earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=substitution}{\\color{@sky}{4x-1=13}}"},{"latex":"\\htmlData{anchor=rearranged}{\\color{@orange}{4x=14}}"},{"latex":"\\htmlData{anchor=xvalue}{\\color{@green}{x=\\frac{7}{2}}}"},{"latex":"\\boxed{\\color{@green}{x=\\frac{7}{2},\\quad y=1}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"substitution"},{"color":"green","style":"text-color-with-background","anchor":"xvalue"}],"connections":[{"to":"rearranged","from":"substitution","color":"orange","label":"add 1"},{"to":"xvalue","from":"rearranged","color":"green","label":"divide by 4"}]},"order":5,"title":"Find x","description":"Substitute y = 1 into the simpler original equation, 4x - y = 13. This earns the A1 mark; state both values together."}]}],"generatedAt":"2026-07-21T04:03:59.062Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"bcbb4573-39b1-4a1a-b79b-c690ec56dea3","revisionId":"326227d4-851a-4935-9863-9f44ae410259","questionNumber":"8","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Simultaneous Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"72d9e156-dd6b-4a63-b09f-3f0182a23365","label":"0","content":"Solve the simultaneous equations\n\n$$3x + y = 15$$\n\n$$5x + 2y = 24$$","markscheme":"- Multiplies first equation by 2: $6x + 2y = 30$ M1\n- Subtracts equations to find $x = 6$ M1\n- Substitutes $x = 6$ to find $y = -3$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Multiplies first equation by 2 to equate coefficients of $y$.","code":"M"},{"id":"p2","marks":1,"description":"Subtracts equations to find $x = 6$.","code":"M"},{"id":"p3","marks":1,"description":"Correct values: $x = 6$ and $y = -3$.","code":"A"}],"finalAnswers":["x=6, y=-3"],"generatedAt":"2026-05-07T01:58:24.365Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"elimination","label":"Elimination","steps":[{"type":"text","order":0,"title":"Match the y terms","content":"Use **elimination**: make the coefficient of $y$ the same in both equations, so that subtracting will remove $y$. The second equation already has $2y$, so multiply every term in the first equation by $2$.","highlights":[{"text":"$$$3x + y = 15$$","location":"specification"},{"text":"$$$5x + 2y = 24$$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{3x+y=15}"},{"latex":"\\htmlData{anchor=doubled}{6x+2y=30}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"doubled"}],"connections":[{"to":"doubled","from":"original","color":"sky","label":"multiply by 2"}]},"order":1,"title":"Multiply the first equation","description":"Multiply every term in the first equation by 2. This earns the first method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=subtraction}{(6x+2y)-(5x+2y)=30-24}"},{"latex":"\\htmlData{anchor=xvalue}{x=6}"}],"highlights":[{"color":"orange","style":"text-color","anchor":"xvalue"}],"connections":[{"to":"xvalue","from":"subtraction","color":"orange","label":"subtract and simplify"}]},"order":2,"title":"Subtract the equations","description":"Subtract the original second equation from the new equation. The identical 2y terms cancel, leaving an equation in x only. This earns the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=substitution}{3(6)+y=15}"},{"latex":"\\htmlData{anchor=simplified}{18+y=15}"},{"latex":"\\htmlData{anchor=solution}{y=-3}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"solution"}],"connections":[{"to":"simplified","from":"substitution","color":"green","label":"simplify"},{"to":"solution","from":"simplified","color":"green","label":"subtract 18"}]},"order":3,"title":"Find y","description":"Put x = 6 into the simpler first original equation, then solve for y. This earns the accuracy mark (A1). Therefore, x = 6 and y = -3."}]}],"generatedAt":"2026-07-21T04:06:02.313Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths","summaries":[{"id":"8147d4f3-4573-4d0b-be74-6fadb7f7af46","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b36b8b68-a092-460f-baf0-55bb8d4de637","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"ae7b0e73-3deb-4449-ba25-2daa95261fe6","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a8fe2b10-0270-4cd4-abb1-7f192ef55745","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"5fa8c900-e93a-47fc-9d15-98beb1042d51","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"843797e3-c08d-4d49-ba9e-c9c84d6b1d47","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"26393c00-9646-48ae-9f1f-7cedf8bec494","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"bcbb4573-39b1-4a1a-b79b-c690ec56dea3","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c78a11c0-af48-4f4f-a023-23dfdbcb16df","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0c738364-9c49-4262-afb5-e192b3685b8c","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d6c4e2bd-e7e0-4b82-8177-71882f13dc11","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6586e07e-4619-4d92-9b2a-e1823a0a8881","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d79a855f-35b0-4930-9776-902f29e2b7a4","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"13d8f2ec-aa95-4f96-9fca-b24a0f1853f0","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true}],"title":"Simultaneous Equations","topicId":"b115e98e-72c2-4592-aaa8-07de09996ab9","topicSidebar":"$L3a","topicSlug":"grade-5-5-8-simultaneous-equations","topicTitle":"Simultaneous Equations"}]}],"$L3b","$L3c"]}]