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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. This earns the accuracy mark."}]}],"generatedAt":"2026-07-21T03:13:49.875Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"d6c4e2bd-e7e0-4b82-8177-71882f13dc11","revisionId":"66c28412-0a51-4025-a73a-057c2f386fbc","questionNumber":"11","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Simultaneous Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"5ba7f8ef-2e21-446b-8a30-013d2b7b9ac6","label":"0","content":"Solve the simultaneous equations\n\n$$\\begin{aligned} 3x &= 9 + y \\\\ x + 5y &= 5 \\end{aligned}$$","markscheme":"- Rearranges first equation to $3x - y = 9$ M1\n- Multiplies second equation by 3 and subtracts to find $y = \\frac{3}{8}$ M1\n- Substitutes $y$ to find $x = \\frac{25}{8}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Rearranges equations into standard form $ax + by = c$.","code":"M"},{"id":"p2","marks":1,"description":"Eliminates a variable and finds the value of $y$ or $x$.","code":"M"},{"id":"p3","marks":1,"description":"Correct values: $x = 3.125$ and $y = 0.375$.","code":"A"}],"finalAnswers":["x=25/8, y=3/8"],"generatedAt":"2026-05-07T01:58:24.365Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"elimination","label":"Elimination","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"3x=9+\\htmlData{anchor=y_initial}{\\color{@sky}{y}}"},{"latex":"3x-\\htmlData{anchor=y_subtracted}{\\color{@sky}{y}}=9"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"y_initial"},{"color":"sky","style":"text-color-with-background","anchor":"y_subtracted"}],"connections":[{"to":"y_subtracted","from":"y_initial","color":"sky","label":"subtract y"}]},"order":1,"title":"Rearrange the first equation","description":"Subtract y from both sides of the first equation. M1: Rearranging gives 3x - y = 9."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=second_eq}{x+5y=5}"},{"latex":"\\htmlData{anchor=multiplied_eq}{3x+15y=15}"},{"latex":"\\left(\\htmlData{anchor=subtract_x_a}{\\color{@sky}{3x}}+15y\\right)-\\left(\\htmlData{anchor=subtract_x_b}{\\color{@sky}{3x}}-y\\right)=15-9"},{"latex":"\\htmlData{anchor=y_sixteen}{16y=6}"},{"latex":"\\htmlData{anchor=y_answer}{\\color{@green}{y=\\frac{3}{8}}}"}],"highlights":[{"color":"green","style":"text-color","anchor":"y_answer"},{"color":"sky","style":"text-color","anchor":"subtract_x_a"},{"color":"sky","style":"text-color","anchor":"subtract_x_b"}],"connections":[{"to":"multiplied_eq","from":"second_eq","color":"orange","label":"multiply by 3"},{"to":"subtract_x_a","from":"multiplied_eq","color":"sky","label":"subtract first equation"},{"to":"y_answer","from":"y_sixteen","color":"green","label":"divide by 16"}]},"order":3,"title":"Eliminate x","description":"Multiply the second equation by 3, then subtract the first equation. The matching 3x terms cancel. M1: Multiply the second equation by 3 and subtract the first to get y = 3/8."},{"type":"hyper_latex","block":{"nodes":[{"latex":"3x-\\htmlData{anchor=three_eighths_sub}{\\color{@sky}{\\frac{3}{8}}}=9"},{"latex":"3x=9+\\htmlData{anchor=three_eighths_add}{\\color{@sky}{\\frac{3}{8}}}"},{"latex":"\\htmlData{anchor=three_x_result}{\\color{@green}{3x}}=\\frac{75}{8}"},{"latex":"\\htmlData{anchor=x_answer}{\\color{@green}{x=\\frac{25}{8}}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"three_eighths_sub"},{"color":"sky","style":"text-color-with-background","anchor":"three_eighths_add"},{"color":"green","style":"text-color-with-background","anchor":"three_x_result"},{"color":"green","style":"text-color-with-background","anchor":"x_answer"}],"connections":[{"to":"three_eighths_add","from":"three_eighths_sub","color":"sky","label":"add 3/8"},{"to":"x_answer","from":"three_x_result","color":"green","label":"divide by 3"}]},"order":5,"title":"Find the value of x","description":"Substitute 3/8 for y in the rearranged equation, then solve for x. A1: Substitute y = 3/8 to get x = 25/8."}]}],"generatedAt":"2026-07-21T04:18:14.469Z","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"]}]