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Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"71fa8ba5-0d49-47c5-bbd5-e51647263be0","label":"0","content":"Solve the simultaneous equations\n\n$$2x + 5y = -10$$\n\n$$2x - y = 8$$","markscheme":"- Subtracts equations to eliminate $x$: $6y = -18$ M1\n- Finds $y = -3$ and substitutes to find $x$ M1\n- Final answers $x = \\frac{5}{2}$ and $y = -3$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Subtracts equations to eliminate $x$.","code":"M"},{"id":"p2","marks":1,"description":"Finds $y = -3$ and substitutes into an equation.","code":"M"},{"id":"p3","marks":1,"description":"Correct values: $x = 2.5$ and $y = -3$.","code":"A"}],"finalAnswers":["x=2.5, 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":"Choose elimination","content":"The coefficient of $x$ is the same in both equations, so subtract the second equation from the first. This eliminates $x$ immediately.","highlights":[{"text":"Solve the simultaneous equations","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=eq1}{\\color{@sky}{2x+5y=-10}}$"},{"latex":"$$\\htmlData{anchor=eq2}{\\color{@amber}{2x-y=8}}$"},{"latex":"$$\\htmlData{anchor=subtract}{\\color{@green}{(2x+5y)-(2x-y)=-10-8}}$"},{"latex":"$$\\htmlData{anchor=result}{\\color{@green}{6y=-18}}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"subtract"},{"color":"green","style":"text-color-with-background","anchor":"result"}],"connections":[{"to":"subtract","from":"eq1","color":"sky","label":"first equation"},{"to":"subtract","from":"eq2","color":"amber","label":"subtract second"},{"to":"result","from":"subtract","color":"green","label":"simplify"}]},"order":1,"title":"Subtract the equations","description":"Subtract every term in the second equation from the corresponding term in the first. Take care: subtracting negative y gives positive y. Reaching 6y = −18 earns the first method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=start}{6y=-18}$"},{"latex":"$$\\htmlData{anchor=divide}{y=\\dfrac{-18}{6}}$"},{"latex":"$$\\htmlData{anchor=yvalue}{\\color{@green}{y=-3}}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"yvalue"}],"connections":[{"to":"divide","from":"start","color":"orange","label":"divide by 6"},{"to":"yvalue","from":"divide","color":"green","label":"simplify"}]},"order":2,"title":"Find y","description":"Divide both sides by 6. This gives y = −3, ready to substitute into one of the original equations."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=original}{2x-y=8}$"},{"latex":"$$\\htmlData{anchor=substitution}{2x-(-3)=8}$"},{"latex":"$$\\htmlData{anchor=simplified}{2x=5}$"},{"latex":"$$\\htmlData{anchor=xvalue}{\\color{@green}{x=\\dfrac{5}{2}}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"substitution"},{"color":"green","style":"text-color-with-background","anchor":"xvalue"}],"connections":[{"to":"substitution","from":"original","color":"sky","label":"substitute y = −3"},{"to":"simplified","from":"substitution","color":"orange","label":"simplify"},{"to":"xvalue","from":"simplified","color":"green","label":"divide by 2"}]},"order":3,"title":"Substitute to find x","description":"Use y = −3 in the simpler second equation. Finding y = −3 and then correctly substituting it to find x earns the second method mark, M1."},{"type":"text","order":4,"title":"State both answers","content":"Write both values clearly:\n\n$$x=\\frac{5}{2},\\qquad y=-3$$\n\nThese correct final answers earn A1."}]}],"generatedAt":"2026-07-20T16:45:00.741Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":55},"walkthroughVideo":null},{"id":"a8fe2b10-0270-4cd4-abb1-7f192ef55745","revisionId":"5e384735-05b5-4e00-9809-71334957ed9c","questionNumber":"4","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Simultaneous Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"bb164942-ce2b-4af3-8d29-f46ac2c3fbf2","label":"0","content":"Solve the simultaneous equations\n\n$$4x + 2y = 10$$\n\n$$5x + 3y = 12$$","markscheme":"- Multiplies equations to equate coefficients (e.g., $20x + 10y = 50$ and $20x + 12y = 48$) M1\n- Subtracts to find $y = -1$ M1\n- Substitutes $y = -1$ to find $x = 3$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Multiplies both equations to equate coefficients of $x$ or $y$.","code":"M"},{"id":"p2","marks":1,"description":"Eliminates a variable and finds the value of $x$ or $y$.","code":"M"},{"id":"p3","marks":1,"description":"Correct values: $x = 3$ and $y = -1$.","code":"A"}],"finalAnswers":["x=3, y=-1"],"generatedAt":"2026-05-07T01:58:24.365Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"elimination","label":"Elimination","steps":[{"type":"text","order":0,"title":"Choose what to eliminate","content":"Eliminate $x$ by making its coefficients equal. The lowest common multiple of $4$ and $5$ is $20$, so multiply the first equation by $5$ and the second equation by $4$.","highlights":[{"text":"Solve the simultaneous equations","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=eq1}{4x+2y=10}$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=scaled1}{20x+10y=50}}$"},{"latex":"$$\\htmlData{anchor=eq2}{5x+3y=12}$"},{"latex":"$$\\color{@amber}{\\htmlData{anchor=scaled2}{20x+12y=48}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"scaled1"},{"color":"amber","style":"text-color","anchor":"scaled2"}],"connections":[{"to":"scaled1","from":"eq1","color":"sky","label":"multiply by 5"},{"to":"scaled2","from":"eq2","color":"amber","label":"multiply by 4"}]},"order":1,"title":"Match the coefficients","description":"Multiplying every term in each equation keeps it balanced. Producing equal coefficients of x earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=subtraction}{(20x+12y)-(20x+10y)=48-50}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=twoy}{2y=-2}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=yvalue}{y=-1}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"twoy"},{"color":"green","style":"text-color-with-background","anchor":"yvalue"}],"connections":[{"to":"twoy","from":"subtraction","color":"sky","label":"subtract like terms"},{"to":"yvalue","from":"twoy","color":"green","label":"divide by 2"}]},"order":2,"title":"Subtract the equations","description":"Subtract the first new equation from the second. The equal x-terms cancel; completing this subtraction to find y earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=original}{4x+2y=10}$"},{"latex":"$$\\htmlData{anchor=substitute_y}{4x+2(-1)=10}$"},{"latex":"$$4x-2=10$"},{"latex":"$$4x=12$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=x_value}{x=3}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"substitute_y"},{"color":"green","style":"text-color-with-background","anchor":"x_value"}],"connections":[{"to":"substitute_y","from":"original","color":"green","label":"substitute y = -1"}]},"order":3,"title":"Substitute to find x","description":"Put y = -1 into the first original equation. Correct substitution and simplification give the A1 mark."},{"type":"text","order":4,"title":"State the solution","content":"Therefore, the simultaneous solution is $\\boxed{x=3,\\ y=-1}$. A quick check gives $4(3)+2(-1)=10$ and $5(3)+3(-1)=12$, so both original equations are satisfied.","highlights":[{"text":"4x + 2y = 10","location":"specification"},{"text":"5x + 3y = 12","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:48:01.908Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":52},"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","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":"3b545d14-c506-4fed-8b22-aa23fb9e2c12","topicSidebar":"$L3b","topicSlug":"grade-5-5-8-simultaneous-equations","topicTitle":"Simultaneous Equations"}]}],"$L3c","$L3d"]}]