,"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 $-2y = 2$, so $y = -1$ A1\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":"Finds $y = -1$.","code":"A"},{"id":"p3","marks":1,"description":"Finds $x = 3$.","code":"A"}],"finalAnswers":["x=3, y=-1"],"generatedAt":"2026-05-06T11:09:32.979Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"elimination","label":"Elimination","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=first_original}{4x+2y=10}$"},{"text":"Multiply every term by 5."},{"latex":"$$\\color{@sky}{\\htmlData{anchor=first_scaled}{20x+10y=50}}$"},{"latex":"$$\\htmlData{anchor=second_original}{5x+3y=12}$"},{"text":"Multiply every term by 4."},{"latex":"$$\\color{@orange}{\\htmlData{anchor=second_scaled}{20x+12y=48}}$"}]},"order":0,"title":"Match the coefficients","description":"The lowest common multiple of the x-coefficients is 20. Multiply every term in the first equation by 5 and every term in the second equation by 4. This earns the M1 method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(20x+10y)-(20x+12y)=50-48$"},{"latex":"$$\\htmlData{anchor=y_equation}{\\color{@sky}{-2y=2}}$"},{"latex":"$$\\htmlData{anchor=y_value}{\\color{@green}{y=-1}}$"}],"connections":[{"to":"y_value","from":"y_equation","color":"green","label":"divide by −2"}]},"order":1,"title":"Subtract the equations","description":"Subtract the second new equation from the first. The 20x terms cancel. Obtaining −2y = 2 and hence y = −1 earns A1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$4x+2y=10$"},{"latex":"$$4x+2(-1)=10$"},{"latex":"$$\\htmlData{anchor=four_x}{\\color{@sky}{4x=12}}$"},{"latex":"$$\\htmlData{anchor=x_value}{\\color{@green}{x=3}}$"},{"latex":"$$\\boxed{(x,y)=(3,-1)}$"}],"connections":[{"to":"x_value","from":"four_x","color":"green","label":"divide by 4"}]},"order":2,"title":"Substitute for y","description":"Put y = −1 into the first original equation, then solve for x. Reaching x = 3 earns the final A1 mark."}]}],"generatedAt":"2026-07-20T11:05:28.074Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"a4fe0162-6214-4dbe-ae50-2ca85d73f366","revisionId":"affcc24d-ca18-4f81-ac01-eba0de434b98","questionNumber":"5","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"hard","totalMarks":3,"sourceTitle":"Simultaneous Equations","calculatorUse":"non_calc","parts":[{"id":"1c3f22b4-8060-48b2-a960-65aafcaf1886","label":"0","content":"Solve the simultaneous equations\n\n$$2x + 5y = 4$$\n\n$$7x - 5y = -1$$","markscheme":"- Adds equations to get $9x = 3$ M1\n- Solves for $x$ to get $x = \\frac{1}{3}$ A1\n- Substitutes $x = \\frac{1}{3}$ to find $y = \\frac{2}{3}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Adds the equations to eliminate $y$.","code":"M"},{"id":"p2","marks":1,"description":"Finds $x = \\frac{1}{3}$.","code":"A"},{"id":"p3","marks":1,"description":"Finds $y = \\frac{2}{3}$.","code":"A"}],"finalAnswers":["x=1/3, y=2/3"],"generatedAt":"2026-05-06T11:09:32.979Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"elimination","label":"Elimination by addition","steps":[{"type":"text","order":0,"title":"Choose to add","content":"The $y$-terms are $+5y$ and $-5y$, so adding the equations eliminates $y$. This is the quickest route because the coefficients are already opposites.","highlights":[{"text":"Solve the simultaneous equations","location":"specification"}]},{"type":"hyper_latex","block":{"no

Simultaneous Equations Questions

Simultaneous Equations Questions

Simultaneous Equations Questions

Simultaneous Equations Questions