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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. 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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},{"id":"c78a11c0-af48-4f4f-a023-23dfdbcb16df","revisionId":"48de04eb-f88c-4e1c-a682-c37e9c36a1c7","questionNumber":"9","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Simultaneous Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"217dcd4f-d3dc-4540-b0f4-8418320a17e0","label":"0","content":"Solve the simultaneous equations\n\n$$3x - y = -4$$\n\n$$2x - 3y = 9$$","markscheme":"- Multiplies first equation by 3: $9x - 3y = -12$ M1\n- Subtracts equations to find $x = -3$ M1\n- Substitutes $x = -3$ to find $y = -5$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Multiplies first equation by 3 to equate coefficients of $y$.","code":"M"},{"id":"p2","marks":1,"description":"Subtracts equations to find $x = -3$.","code":"M"},{"id":"p3","marks":1,"description":"Correct values: $x = -3$ and $y = -5$.","code":"A"}],"finalAnswers":["x=-3, y=-5"],"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 coefficients of one letter the same, so that letter disappears when the equations are subtracted. The $y$-terms are $-y$ and $-3y$, so multiply the first equation by $3$.","highlights":[{"text":"Solve the simultaneous equations","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{\\color{@sky}{3x-y=-4}}"},{"latex":"\\htmlData{anchor=multiplied}{\\color{@orange}{9x-3y=-12}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"original"},{"color":"orange","style":"text-color-with-background","anchor":"multiplied"}],"connections":[{"to":"multiplied","from":"original","color":"orange","label":"multiply by 3","style":"solid"}]},"order":1,"title":"Multiply the first equation","description":"Multiply every term in the first equation by 3. This earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=first}{\\color{@sky}{9x-3y=-12}}"},{"latex":"\\htmlData{anchor=second}{\\color{@orange}{2x-3y=9}}"},{"latex":"\\htmlData{anchor=operation}{\\left(9x-3y\\right)-\\left(2x-3y\\right)=-12-9}"},{"latex":"\\htmlData{anchor=result}{\\color{@amber}{7x=-21}}"},{"latex":"\\htmlData{anchor=xvalue}{\\color{@green}{x=-3}}"}],"highlights":[{"color":"amber","style":"text-color-with-background","anchor":"result"}],"connections":[{"to":"operation","from":"first","color":"sky","label":"first equation","style":"solid"},{"to":"operation","from":"second","color":"orange","style":"dashed"},{"to":"result","from":"operation","color":"amber","label":"simplify","style":"solid"},{"to":"xvalue","from":"result","color":"green","label":"divide by 7","style":"solid"}]},"order":2,"title":"Subtract the equations","description":"Subtract the second equation from the first. The negative three y terms cancel, leaving an equation in x only; this earns the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=substitution}{3\\left(\\color{@sky}{-3}\\right)-y=-4}"},{"latex":"\\htmlData{anchor=evaluated}{-9-y=-4}"},{"latex":"\\htmlData{anchor=minusy}{-y=5}"},{"latex":"\\htmlData{anchor=yvalue}{\\color{@green}{y=-5}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"substitution"},{"color":"green","style":"text-color-with-background","anchor":"yvalue"}],"connections":[{"to":"evaluated","from":"substitution","color":"sky","label":"evaluate 3(-3)","style":"solid"},{"to":"minusy","from":"evaluated","color":"amber","label":"add 9","style":"solid"},{"to":"yvalue","from":"minusy","color":"green","label":"multiply by -1","style":"solid"}]},"order":3,"title":"Find y","description":"Put x = -3 into the first equation, then solve for y. 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