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A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correctly collects terms on either side, e.g., $11x - 4x = 23 - 5$ or $7x = 18$","code":"M"},{"id":"p2","marks":1,"description":"Correct final answer as a fraction or decimal","code":"A"}],"finalAnswers":["\\frac{18}{7}","2.57"],"generatedAt":"2026-05-06T14:00:06.788Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"balance_equation","label":"Balance the equation","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=collect_start}{\\color{@sky}{11x-4x=23-5}}$"},{"latex":"$$\\htmlData{anchor=collected}{\\color{@green}{7x=18}}$"}],"connections":[{"to":"collected","from":"collect_start","color":"sky","style":"solid"}]},"order":1,"title":"Collect like terms","description":"Subtract 4x from both sides and add 23 to both sides. Do the same operation to both sides to keep the equation balanced. The result 7x = 18 earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=divide_start}{\\color{@sky}{7x=18}}$"},{"latex":"$$\\htmlData{anchor=divided}{\\color{@orange}{\\dfrac{7x}{7}=\\dfrac{18}{7}}}$"},{"latex":"$$\\htmlData{anchor=answer}{\\color{@green}{x=\\dfrac{18}{7}}}$"}],"connections":[{"to":"divided","from":"divide_start","color":"sky","style":"solid"},{"to":"answer","from":"divided","color":"green","style":"solid"}]},"order":2,"title":"Divide by seven","description":"Divide both sides by 7, then cancel the 7 on the left to leave x on its own. The exact fraction 18/7 earns A1; 2.57 to 2 decimal places is also accepted."}]}],"generatedAt":"2026-07-19T18:33:21.833Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":69},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/e59166a6-ed31-4da2-993c-7092b59da8d2.mp4","durationSeconds":53,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/e59166a6-ed31-4da2-993c-7092b59da8d2.jpg"}},{"id":"b3845a4f-8a64-4233-98b9-cffd5b3e9847","revisionId":"85ba68ee-75c4-4b37-89f0-b7c9efd97c19","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"hard","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"6fa7a5e2-8eea-4ea0-9c6e-11b923b5380b","label":"a","content":"Solve $12b - 5 = 8b + 19$","markscheme":"- Collects like terms: $4b = 24$ or $12b - 8b = 19 + 5$ M1\n- Final answer 6 A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correctly grouping the variable terms and numerical terms on opposite sides.","code":"M"},{"id":"p2","marks":1,"description":"Correct final value for $b$.","code":"A"}],"finalAnswers":["6"],"generatedAt":"2026-05-06T14:00:10.609Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"collect_like_terms","label":"Collect like terms","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=original}{\\color{@sky}{12b-5=8b+19}}$"},{"latex":"$$\\htmlData{anchor=collected}{\\color{@sky}{12b-8b=19+5}}$"},{"latex":"$$\\htmlData{anchor=simplified}{\\color{@green}{4b=24}}$"}],"connections":[{"to":"collected","from":"original","color":"sky","label":"subtract 8b; add 5"},{"to":"simplified","from":"collected","color":"green","label":"simplify"}]},"order":1,"title":"Collect like terms","description":"Subtract 8b from both sides, then add 5 to both sides to keep the equation balanced. Reaching 4b = 24 earns the method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=equation}{\\color{@sky}{4b=24}}$"},{"latex":"$$\\htmlData{anchor=divide}{\\color{@orange}{\\dfrac{4b}{4}=\\dfrac{24}{4}}}$"},{"latex":"$$\\htmlData{anchor=answer}{\\color{@green}{b=6}}$"}],"connections":[{"to":"divide","from":"equation","color":"orange","label":"divide both sides by 4"},{"to":"answer","from":"divide","color":"green","label":"simplify"}]},"order":2,"title":"Divide both sides","description":"Divide both sides by 4 to leave b on its own. The correct final answer earns A1. Optionally, substitute b = 6 into the original equation to check that both sides are equal."}]}],"generatedAt":"2026-07-19T06:27:53.098Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"6fa7a5e2-8eea-4ea0-9c6e-11b923b5380b:algebraic_isolation","label":"Algebraic Isolation","steps":[{"type":"text","order":0,"title":"Identify the equation","content":"We start with the given linear equation:\n\n$$12b - 5 = 8b + 19$$\n\nTo solve for $b$, our goal is to isolate the terms involving the variable $b$ on one side of the equation and the constant numbers on the other side.","highlights":[{"text":"Solve 12b - 5 = 8b + 19","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$12b - 5 = \\htmlData{anchor=sub-term}{\\color{@sky}{8b}} + 19$"},{"latex":"$$\\htmlData{anchor=lhs-result}{\\color{@sky}{12b - 8b}} - 5 = 19$"},{"latex":"$$4b - 5 = 19$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-term"},{"color":"sky","style":"text-color","anchor":"lhs-result"}],"connections":[{"to":"lhs-result","from":"sub-term","color":"sky","label":"Subtract 8b","style":"solid"}]},"order":1,"title":"Isolate variable terms","description":"Subtract $8b$ from both sides of the equation to bring all variable terms to the left side."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$4b \\htmlData{anchor=add-term}{\\color{@amber}{- 5}} = 19$"},{"latex":"$$4b = \\htmlData{anchor=rhs-result}{\\color{@amber}{19 + 5}}$"},{"latex":"$$4b = 24$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"add-term"},{"color":"amber","style":"text-color","anchor":"rhs-result"}],"connections":[{"to":"rhs-result","from":"add-term","color":"amber","label":"Add 5","style":"solid"}]},"order":2,"title":"Isolate constant terms","description":"Add $5$ to both sides of the equation to collect all constant numerical terms on the right side. Showing this step earns the first mark M1."},{"type":"text","order":3,"title":"Solve for b","content":"Now we have simplified the equation to:\n\n$$4b = 24$$\n\nTo find the value of $b$, divide both sides by $4$:\n\n$$b = \\frac{24}{4}$$\n\n$$b = 6$$\n\nThis gives us our final answer, which awards the second mark A1."}]},{"id":"6fa7a5e2-8eea-4ea0-9c6e-11b923b5380b:graphical_intersection","label":"Graphical Intersection","steps":[{"type":"text","order":0,"title":"Define two functions","content":"To solve the equation $12b - 5 = 8b + 19$ using a graphical method, we can set each side of the equation equal to $y$. This gives us a system of two linear equations where the independent variable is $b$ (which we can treat as $x$ on a standard graph):\n\n$$y = 12b - 5$$\n$$y = 8b + 19$$","highlights":[{"text":"Solve $12b - 5 = 8b + 19$","location":"specification"}]},{"type":"text","order":1,"title":"Graph the equations","content":"Next, we plot both lines on a coordinate grid where the horizontal axis represents $b$ and the vertical axis represents $y$. \n\n* The line $$y = 12b - 5$$ has a $y$-intercept at $(0, -5)$ and a steep gradient of $12$.\n* The line $$y = 8b + 19$$ has a $y$-intercept at $(0, 19)$ and a slightly gentler gradient of $8$."},{"type":"text","order":2,"title":"Find the intersection","content":"The solution to our equation is the $b$-coordinate of the point where the two lines intersect. Using a graphing tool or calculator, we can find this intersection point directly.","calculator":[{"gdc":{"instructions":"1. Go to the Graph menu.\n2. Enter the functions \\|f1(x) = 12x - 5\\| and \\|f2(x) = 8x + 19\\|.\n3. Adjust the view window (e.g., x from 0 to 10, y from 0 to 80).\n4. Use the Intersection tool (typically G-Solv -> INTSECT or Analyze Graph -> Intersection).\n5. The tool will show the intersection point at (6, 67)."},"ti84":{"instructions":"1. Press [Y=] and enter \\|Y1 = 12X - 5\\| and \\|Y2 = 8X + 19\\|.\n2. Press [WINDOW] and set Xmin = 0, Xmax = 10, Ymin = 0, Ymax = 100 to ensure the intersection is visible.\n3. Press [GRAPH].\n4. Press [2nd] [TRACE] (CALC) and select '5: intersect'.\n5. Press [ENTER] for the First curve, Second curve, and Guess.\n6. The calculator will display the intersection at X = 6 and Y = 67."},"desmos":{"instructions":"1. Type \\|y = 12x - 5\\| in the first expression bar.\n2. Type \\|y = 8x + 19\\| in the second expression bar.\n3. Adjust the graph zoom until you see where the two lines cross.\n4. Click on the gray point where the two lines intersect to reveal the coordinates: (6, 67)."},"description":"Find the point of intersection of the two lines."}]},{"type":"text","order":3,"title":"State final solution","content":"From the coordinates of the intersection point $(6, 67)$, we find:\n\n$$b = 6$$\n\nThis confirms our solution. On a non-calculator exam, this corresponds to rearranging the equation to collect like terms:\n$$\\begin{aligned} 12b - 8b &= 19 + 5 \\\\ 4b &= 24 \\\\ b &= 6 \\end{aligned}$$\n\nBoth methods lead to the final correct 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Solving Equations

Question 3

Solving Equations Questions

Solving Equations

Question 3

Solving Equations Questions