nal A1."},{"type":"text","order":4,"title":"State and check","content":"State the solution as $x=3$ and $y=2$. A quick check gives $5(3)+2=17$ and $3(3)+2(2)=13$, so the pair satisfies both original equations."}]}],"generatedAt":"2026-07-19T07:49:08.879Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"f2b25ee8-2fc1-4565-b566-5f11e9c9123d","revisionId":"3386b89b-9625-4acf-96a8-8f53fee5ccbf","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"hard","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"39b34c98-c77c-45e8-bbfe-abe38101cc73","label":"a","content":"Solve the simultaneous equations\n\n$$10x + 4y = 6$$\n\n$$5x - 3y = 10.5$$","markscheme":"- Multiplies second equation by 2 to get $10x - 6y = 21$ M1\n- Subtracts equations to find $10y = -15$, so $y = -1.5$ A1\n- Substitutes $y = -1.5$ to find $x = 1.2$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Multiplies second equation by 2 to get $10x - 6y = 21$ or equivalent.","code":"M"},{"id":"p2","marks":1,"description":"Subtracts to find $y = -1.5$ (or $x = 1.2$).","code":"A"},{"id":"p3","marks":1,"description":"Substitutes to find $x = 1.2$ (or $y = -1.5$).","code":"A"}],"finalAnswers":["x = 1.2, y = -1.5"],"generatedAt":"2026-05-06T14:06:39.389Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"elimination","label":"Elimination","steps":[{"type":"text","order":0,"title":"Choose what to eliminate","content":"The coefficients of $x$ are $10$ and $5$. Doubling the second equation makes both coefficients $10$, so subtracting the equations will eliminate $x$.","highlights":[{"text":"Solve the simultaneous equations","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$5x-3y=10.5$"},{"latex":"$$10x-6y=21$"}]},"order":1,"title":"Multiply the equation","description":"Multiply every term in the second equation by 2. Writing the correctly scaled equation earns M1, even if a later subtraction slips."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$10x+4y=6$"},{"latex":"$$10x-6y=21$"},{"latex":"$$(10x+4y)-(10x-6y)=6-21$"},{"latex":"$$10y=-15$"},{"latex":"$$y=-1.5$"}]},"order":2,"title":"Subtract the equations","description":"Subtract the scaled second equation from the first. The 10x terms cancel, and 4y − (−6y) becomes 10y. Finding y = −1.5 earns A1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$10x+4y=6$"},{"latex":"$$10x+4(-1.5)=6$"},{"latex":"$$10x-6=6$"},{"latex":"$$10x=12$"},{"latex":"$$x=1.2$"}]},"order":3,"title":"Substitute for y","description":"Put y = −1.5 into either original equation. The correct value x = 1.2 earns the final A1. So the simultaneous solution is x = 1.2, y = −1.5; these values satisfy both original equations."}]}],"generatedAt":"2026-07-19T18:51:40.704Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"f347b742-e77c-4b83-9f92-c20744329631","revisionId":"bdfbaaee-e1bf-4533-933b-7814281d51e1","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"hard","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"e6d2c5de-f9b9-4db7-8792-f80f378172e0","label":"a","content":"Solve the simultaneous equations\n\n$$5x + 3y = 14$$\n\n$$x - 3y = 4$$","markscheme":"- Adds equations to get $6x = 18$ M1\n- Solves for $x$ to get $x = 3$ A1\n- Substitutes $x=3$ to find $y = -\\frac{1}{3}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Method to eliminate $y$ by adding equations together.","code":"M"},{"id":"p2","marks":1,"description":"Correct $x$ value found.","code":"A"},{"id":"p3","marks":1,"description":"Correct $y$ value found.","code":"A"}],"finalAnswers":["x = 3, y = -\\frac{1}{3}","x = 3

Simultaneous Equations Questions

Simultaneous Equations Questions

Simultaneous Equations Questions

Simultaneous Equations Questions