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bg-background","children":["$","$L39",null,{"backHref":"/gcse/maths/edexcel/questions","basePath":"/gcse/maths/edexcel/grade-3-3-14-solving-equations/questions","formulaUrl":"https://assets.mathsgenie.co.uk/board-resources/gcse-maths/gcse-formulae-to-remember.pdf","initialIndex":58,"initialQuestionWindow":[{"id":"018de84b-32d7-49b3-b6d0-606a06b94228","revisionId":"c8354719-c444-49c8-84be-a2012c112b6e","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"hard","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"6dfb31dc-22da-40b3-bd1c-762886dbd804","label":"a","content":"Solve $15 + 2h = 5h - 12$","markscheme":"- Collects like terms: $27 = 3h$ or $15 + 12 = 5h - 2h$ M1\n- Final answer 9 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 $h$.","code":"A"}],"finalAnswers":["9"],"generatedAt":"2026-05-06T14:00:12.487Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":{"methods":[{"id":"algebraic","label":"Algebraic Isolation","steps":[{"type":"text","order":0,"title":"Group the variable terms","content":"To solve for $h$, we first want to collect all terms containing $h$ on one side of the equation. We can do this by subtracting $2h$ from both sides:\n\n$$\\begin{aligned} 15 + 2h &= 5h - 12 \\\\ 15 &= 3h - 12 \\end{aligned}$$\n\nThis keeps the coefficient of $h$ positive, which often makes the calculation easier.","highlights":[{"text":"15 + 2h = 5h - 12","location":"specification"}]},{"type":"text","order":1,"title":"Isolate the constant terms","content":"Now we want to move the constant terms (the numbers without a variable) to the other side. Since $12$ is being subtracted from $3h$, we perform the inverse operation by adding $12$ to both sides:\n\n$$\\begin{aligned} 15 + 12 &= 3h \\\\ 27 &= 3h \\end{aligned}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=val}{\\color{@sky}{27}} = \\htmlData{anchor=coeff}{\\color{@amber}{3}}h$"},{"latex":"$$h = \\frac{\\htmlData{anchor=val-sub}{\\color{@sky}{27}}}{\\htmlData{anchor=coeff-sub}{\\color{@amber}{3}}}$"},{"latex":"$$h = 9$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"val"},{"color":"amber","style":"text-color","anchor":"coeff"}],"connections":[{"to":"val-sub","from":"val","color":"sky","label":null,"style":"solid"},{"to":"coeff-sub","from":"coeff","color":"amber","label":null,"style":"solid"}]},"order":2,"title":"Solve for h","description":"To get $h$ by itself, we divide both sides by its coefficient, which is $3$."},{"type":"text","order":3,"title":"Verify with a calculator","content":"You can check your answer by using a solver or graphing tool to find the intersection of both sides of the equation.","calculator":[{"gdc":{"expression":"x = 9","instructions":"1. Open the Graphing menu.\n2. Enter f1(x) = 15 + 2x and f2(x) = 5x - 12.\n3. Use the 'Intersection' tool (often under Analyze Graph or G-Solve).\n4. Select the two lines."},"ti84":{"expression":"X=9, Y=33","instructions":"1. Press [Y=].\n2. Enter Y1 = 15 + 2X and Y2 = 5X - 12.\n3. Press [GRAPH]. Adjust window if needed (try Xmin=0, Xmax=15, Ymin=0, Ymax=40).\n4. Press [2nd] [TRACE] and select '5: intersect'.\n5. Press [ENTER] three times."},"desmos":{"expression":"(9, 33)","instructions":"1. Type y = 15 + 2x in the first row.\n2. Type y = 5x - 12 in the second row.\n3. Click on the point where the two lines cross."},"description":"Find the value of $h$ by finding the intersection of $y = 15 + 2x$ and $y = 5x - 12$."}]}]},{"id":"graphical","label":"Graphical Approach","steps":[{"type":"text","order":0,"title":"Define the linear functions","content":"To solve the equation using a graphical approach, we treat each side of the equals sign as a separate linear function. Let's use $x$ as our horizontal axis (representing $h$) and $y$ as our vertical axis.\n\n$$\\begin{aligned} y_1 &= 15 + 2x \\\\ y_2 &= 5x - 12 \\end{aligned}$$\n\nThe solution to the equation will be the $x$-coordinate where these two lines intersect.","highlights":[{"text":"15 + 2h = 5h - 12","location":"specification"}]},{"type":"text","order":1,"title":"Graph the functions","content":"Using a graphing calculator or software, we can plot both lines. \n\n- The first line, $y_1 = 2x + 15$, has a $y$-intercept at $15$ and a positive gradient of $2$.\n- The second line, $y_2 = 5x - 12$, starts lower (intercept at $-12$) but is much steeper with a gradient of $5$.","calculator":[{"gdc":{"expression":"f(x) = 2x + 15, g(x) = 5x - 12","instructions":"1. Open the Graphing menu.\n2. Enter the functions $f1(x) = 2x + 15$ and $f2(x) = 5x - 12$.\n3. Use the 'Analyze Graph' or 'G-Solv' tool and select 'Intersection'."},"ti84":{"expression":"y = 2x + 15, y = 5x - 12","instructions":"1. Press [Y=].\n2. Enter `Y1 = 2X + 15` and `Y2 = 5X - 12`.\n3. Press [WINDOW] and set Xmin=0, Xmax=15, Ymin=0, Ymax=40 to see the crossing point.\n4. Press [GRAPH].\n5. Press [2nd] [TRACE] then select '5: intersect'.\n6. Press [ENTER] three times to find the point."},"desmos":{"expression":"y = 2x + 15, y = 5x - 12","instructions":"1. Type `y = 2x + 15` in the first row.\n2. Type `y = 5x - 12` in the second row.\n3. Click on the gray point where the two lines cross to reveal the coordinates."},"description":"Graph the two equations and find their point of intersection."}]},{"type":"hyper_latex","block":{"nodes":[{"text":"The intersection point is:"},{"latex":"$$\\htmlData{anchor=intersect}{\\color{@sky}{(9, 33)}}$"},{"text":"This means when $h = 9$, both sides equal 33."},{"latex":"$$\\htmlData{anchor=solution}{\\color{@sky}{h = 9}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"intersect"}],"connections":[{"to":"solution","from":"intersect","color":"sky","label":"x-coordinate","style":"solid"}]},"order":2,"title":"Identify the intersection","description":"The point where the lines cross tells us the value of $x$ (which is our $h$) that makes both sides of the equation equal."},{"type":"text","order":3,"title":"State the final answer","content":"Since the $x$-coordinate of the intersection is $9$, the value of $h$ that satisfies the equation is $9$. \n\nYou can check this by substituting back: \n$$15 + 2(9) = 33$$\n$$5(9) - 12 = 33$$\nBoth sides match perfectly!"}]}],"generatedAt":"2026-06-08T01:46:41.232Z","modelVersion":"gemini-3-flash-preview","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":70},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/018de84b-32d7-49b3-b6d0-606a06b94228.mp4","durationSeconds":55,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/018de84b-32d7-49b3-b6d0-606a06b94228.jpg"}},{"id":"05bbeb3c-361e-4f76-9241-13241928887c","revisionId":"4af43a67-21c2-4908-8444-e4630be508f8","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"hard","totalMarks":4,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"32de4619-af8f-46cc-b21a-a8ad9031a6a1","label":"a","content":"Solve $3m + 48 = 11m$","markscheme":"- Subtracts $3m$ from both sides: $48 = 8m$ M1\n- Final answer 6 A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Subtracts 3m from both sides to obtain 48 = 8m","code":"M"},{"id":"p2","marks":1,"description":"Correct final answer of 6","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T14:00:17.181Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"collect_terms","label":"Collect the variable terms","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3m+48=11m$"},{"latex":"$$\\htmlData{anchor=subtract_line}{3m-\\color{@sky}{3m}+48=11m-\\color{@sky}{3m}}$"},{"latex":"$$\\htmlData{anchor=collected}{48=\\color{@orange}{8m}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"subtract_line"},{"color":"orange","style":"text-color","anchor":"collected"}],"connections":[{"to":"collected","from":"subtract_line","color":"orange","label":"collect like terms"}]},"order":1,"title":"Subtract from both sides","description":"Subtract 3m from both sides to keep the equation balanced. This gives 48 = 8m and earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$48=\\htmlData{anchor=eight_m}{\\color{@orange}{8m}}$"},{"latex":"$$\\htmlData{anchor=divide}{\\dfrac{48}{8}=\\dfrac{8m}{8}}$"},{"latex":"$$\\htmlData{anchor=answer}{m=\\color{@green}{6}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"eight_m"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"divide","from":"eight_m","color":"orange","label":"divide both sides by 8"},{"to":"answer","from":"divide","color":"green","label":"simplify"}]},"order":2,"title":"Divide by eight","description":"Divide both sides by 8 to leave m on its own. The answer m = 6 earns A1."}]}],"generatedAt":"2026-07-19T07:10:31.400Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"10f777cd-a885-4ef9-bff6-c102c2821c11","label":"b","content":"Solve $48 = 8(4t - 2)$","markscheme":"- Expands brackets or divides by 8: $48 = 32t - 16$ or $6 = 4t - 2$ M1\n- Final answer 2 A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correct expansion of brackets (32t - 16) or correct division of both sides by 8","code":"M"},{"id":"p2","marks":1,"description":"Correct final answer of 2","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T14:00:17.181Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":{"methods":[{"id":"algebraic_isolation","label":"Algebraic Isolation","steps":[{"type":"text","order":0,"title":"Group variable terms","content":"To solve for $m$, we need to get all the terms with $m$ on one side of the equation and the constant numbers on the other. We can start by subtracting $3m$ from both sides of the equation to isolate the constant $48$.","highlights":[{"text":"3m + 48 = 11m","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=start-lhs}{3m} + 48 = \\htmlData{anchor=start-rhs}{11m}$"},{"latex":"$$48 = \\htmlData{anchor=calc-rhs}{11m - 3m}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"start-lhs"},{"color":"sky","style":"text-color","anchor":"calc-rhs"}],"connections":[{"to":"calc-rhs","from":"start-lhs","color":"sky","label":"subtract 3m","style":"solid"}]},"order":1,"title":"Subtract 3m from both sides","description":"Subtracting $3m$ from the left side leaves only the number 48, while subtracting it from the right side combines the $m$ terms."},{"type":"text","order":2,"title":"Simplify the equation","content":"Now, simplify the right side by calculating $11m - 3m$:\n\n$$48 = 8m$$\n\nThis step earns you the first mark (M1) for correctly rearranging the equation to isolate the variable terms."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=coeff}{8}m = \\htmlData{anchor=const}{48}$"},{"latex":"$$m = \\frac{\\htmlData{anchor=const-top}{48}}{\\htmlData{anchor=coeff-bot}{8}}$"},{"latex":"$$m = 6$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"coeff"},{"color":"sky","style":"text-color","anchor":"const"}],"connections":[{"to":"coeff-bot","from":"coeff","color":"amber","label":null,"style":"solid"},{"to":"const-top","from":"const","color":"sky","label":null,"style":"solid"}]},"order":3,"title":"Solve for m","description":"To find the value of $m$, divide both sides by its coefficient, 8."},{"type":"text","order":4,"title":"State the final answer","content":"The solution is $m = 6$. You can check this by substituting it back into the original equation: $3(6) + 48 = 18 + 48 = 66$, and $11(6) = 66$. Both sides match!","calculator":[{"gdc":{"expression":"3x + 48 = 11x","instructions":"Go to the Equation menu, select 'Solver'. Enter '3x + 48 = 11x' and solve for x."},"ti84":{"expression":"3X + 48 = 11X","instructions":"Press [MATH], scroll to 'Solver...' or 'Numeric Solver'. Enter 'E1: 3X + 48' and 'E2: 11X'. Press [OK] (Graph), then [ALPHA] [SOLVE]."},"desmos":{"expression":"3m + 48 = 11m","instructions":"Type the equation into the first row. Desmos will show a vertical line at the value of the solution."},"description":"You can solve this equation numerically or by finding the intersection of two lines."}]}]},{"id":"trial_and_improvement","label":"Trial and Improvement","steps":[{"type":"text","order":0,"title":"Define the goal","content":"To solve the equation using **Trial and Improvement**, we will pick different values for $m$ and substitute them into both sides of the equation until we find a value that makes both sides equal.\n\nThe equation is:\n$$3m + 48 = 11m$$\n\nWe want to find $m$ such that the left-hand side (LHS) is the same as the right-hand side (RHS).","highlights":[{"text":"3m + 48 = 11m","location":"specification"}]},{"type":"text","order":1,"title":"Try an initial value","content":"Let's start by trying a small value, like $m = 1$.\n\n- **LHS:** $3(1) + 48 = 3 + 48 = 51$\n- **RHS:** $11(1) = 11$\n\nSince $51$ is much larger than $11$, we need to try a much larger value for $m$.","calculator":[{"gdc":{"expression":"3*(1)+48\n11*(1)","instructions":"Calculate the value for each side of the equation."},"ti84":{"expression":"3*(1)+48\n11*(1)","instructions":"Enter the expressions using 1 as the variable."},"desmos":{"expression":"3(1)+48\n11(1)","instructions":"Type the expressions into separate lines to compare values."},"description":"Evaluate the expressions for m = 1"}]},{"type":"text","order":2,"title":"Refine the guess","content":"Let's try a larger value, such as $m = 5$, to see if the sides get closer.\n\n- **LHS:** $3(5) + 48 = 15 + 48 = 63$\n- **RHS:** $11(5) = 55$\n\nThey are much closer now ($63$ and $55$), but the LHS is still slightly larger. Let's try the next whole number.","calculator":[{"gdc":{"expression":"3*(5)+48\n11*(5)","instructions":"Calculate the value for each side of the equation."},"ti84":{"expression":"3*(5)+48\n11*(5)","instructions":"Enter the expressions using 5 as the variable."},"desmos":{"expression":"3(5)+48\n11(5)","instructions":"Type the expressions into separate lines to compare values."},"description":"Evaluate the expressions for m = 5"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$m = \\htmlData{anchor=test-val}{\\color{@sky}{6}}$"},{"latex":"\\text{LHS: } 3(\\htmlData{anchor=sub1}{\\color{@sky}{6}}) + 48 = \\htmlData{anchor=res1}{18 + 48 = 66}"},{"latex":"\\text{RHS: } 11(\\htmlData{anchor=sub2}{\\color{@sky}{6}}) = \\htmlData{anchor=res2}{66}"}],"highlights":[{"color":"green","style":"box","anchor":"res1"},{"color":"green","style":"box","anchor":"res2"}],"connections":[{"to":"sub1","from":"test-val","color":"sky","label":null,"style":"solid"},{"to":"sub2","from":"test-val","color":"sky","label":null,"style":"solid"}]},"order":3,"title":"Test the next value","description":"Now we test $m = 6$ to see if it results in equality."},{"type":"text","order":4,"title":"State final answer","content":"Since both the left-hand side and the right-hand side equal $66$ when $m = 6$, we have found the solution.\n\n$$m = 6$$"}]},{"id":"expansion","label":"Expansion of Brackets","steps":[{"type":"text","order":0,"title":"Identify the equation","content":"We need to solve the equation $48 = 8(4t - 2)$ for the variable $t$. Our approach will be to expand the brackets first.","highlights":[{"text":"Solve $48 = 8(4t - 2)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$48 = \\htmlData{anchor=outer}{\\color{@sky}{8}}(\\htmlData{anchor=term1}{\\color{@amber}{4t}} \\htmlData{anchor=term2}{\\color{@purple}{- 2}})$"},{"latex":"$$48 = (\\color{@sky}{8} \\times \\color{@amber}{4t}) + (\\color{@sky}{8} \\times \\color{@purple}{-2})$"},{"latex":"$$\\htmlData{anchor=result}{48 = 32t - 16}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"outer"},{"color":"amber","style":"text-color","anchor":"term1"},{"color":"purple","style":"text-color","anchor":"term2"}],"connections":[{"to":"result","from":"outer","color":"sky","label":"distribute","style":"solid"}]},"order":1,"title":"Expand the brackets","description":"Using the distributive property, multiply the number outside the brackets by each term inside."},{"type":"text","order":2,"title":"Isolate the variable term","content":"To get the term with $t$ by itself, we need to move the $-16$ to the other side by adding $16$ to both sides of the equation:\n\n$$\\begin{aligned} 48 + 16 &= 32t \\\\ 64 &= 32t \\end{aligned}$$"},{"type":"text","order":3,"title":"Solve for t","content":"Now, divide both sides by $32$ to find the value of $t$:\n\n$$\\begin{aligned} t &= \\frac{64}{32} \\\\ t &= 2 \\end{aligned}$$\n\nYou would earn one mark (M1) for correctly expanding the brackets to get $48 = 32t - 16$, and the second mark (A1) for the final answer.","calculator":[{"gdc":{"expression":"y=48, y=8(4x-2)","instructions":"Go to the Graphing menu. Enter f1(x)=48 and f2(x)=8(4x-2). Use the G-Solve tool to find the 'Intersection'."},"ti84":{"expression":"Y1=48, Y2=8(4x-2)","instructions":"Press [Y=] and enter Y1=48 and Y2=8(4X-2). Press [GRAPH]. Use [2nd] [CALC] [5:intersect] to find where they cross."},"desmos":{"expression":"y=48, y=8(4x-2)","instructions":"Type y=48 in the first row and y=8(4x-2) in the second row. Click on the point where the two lines cross to see the x-coordinate."},"description":"You can verify this result by graphing both sides of the equation and finding the intersection."}]}]},{"id":"division","label":"Initial Division","steps":[{"type":"text","order":0,"title":"Divide by the common factor","content":"To solve the equation $48 = 8(4t - 2)$, we can start by simplifying the right-hand side. Since the entire bracket is multiplied by 8, let's divide both sides of the equation by 8 first.\n\n$$\\frac{48}{8} = 4t - 2$$\n\nCalculating the left side gives us:\n\n$$6 = 4t - 2$$","calculator":[{"gdc":{"expression":"48 / 8","instructions":"Type 48 / 8 in the calculation screen."},"ti84":{"expression":"48 / 8","instructions":"Type 48 divided by 8 and press ENTER."},"desmos":{"expression":"48 / 8","instructions":"Type 48 / 8 into a cell."},"description":"You can verify the initial division using your calculator."}],"highlights":[{"text":"48 = 8(4t - 2)","location":"specification"}]},{"type":"text","order":1,"title":"Isolate the variable term","content":"Now we want to get the term with $t$ by itself. We do this by adding 2 to both sides of the equation:\n\n$$\\begin{aligned} 6 + 2 &= 4t - 2 + 2 \\\\ 8 &= 4t \\end{aligned}$$"},{"type":"text","order":2,"title":"Solve for t","content":"Finally, to find the value of $t$, divide both sides by 4:\n\n$$\\begin{aligned} \\frac{8}{4} &= \\frac{4t}{4} \\\\ 2 &= t \\end{aligned}$$\n\nSo, the final answer is $t = 2$.","calculator":[{"gdc":{"instructions":"Graph f1(x) = 48 and f2(x) = 8(4x - 2). Use the intersection tool to find the x-coordinate."},"ti84":{"instructions":"Go to Y=, set Y1 = 48 and Y2 = 8(4X - 2). Use 2nd -> TRACE -> intersect to find where they meet."},"desmos":{"expression":"y = 48, y = 8(4x - 2)","instructions":"Type y = 48 and y = 8(4x - 2). Click on the point of intersection to see the x-value."},"description":"You can check your answer by graphing both sides of the original 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