l}{\\color{@sky}{m}}(x - \\htmlData{anchor=x1}{\\color{@purple}{x_1}})$$"},{"text":"Using our known point $P(\\color{@purple}{2}, \\color{@red}{-5})$ and gradient $m = \\color{@sky}{\\frac{2}{3}}$:"},{"latex":"$$$y - (\\htmlData{anchor=y1-sub}{\\color{@red}{-5}}) = \\htmlData{anchor=m-sub}{\\color{@sky}{\\frac{2}{3}}}(x - \\htmlData{anchor=x1-sub}{\\color{@purple}{2}})$$"},{"text":"Simplifying the left-hand side gives:"},{"latex":"$$$y + 5 = \\frac{2}{3}(x - 2)$$"}],"highlights":[{"color":"red","style":"text-color","anchor":"y1"},{"color":"purple","style":"text-color","anchor":"x1"},{"color":"sky","style":"text-color","anchor":"m-val"}],"connections":[{"to":"y1-sub","from":"y1","color":"red","label":"y-coordinate of P","style":"solid"},{"to":"x1-sub","from":"x1","color":"purple","label":"x-coordinate of P","style":"solid"},{"to":"m-sub","from":"m-val","color":"sky","label":"gradient","style":"solid"}]},"order":1,"title":"Find the line's equation","description":"We establish the general linear equation using point P and substitute our values."},{"type":"text","order":2,"title":"Substitute the second point","content":"Since point $Q(8, k)$ lies on this line, its coordinates must satisfy the equation. \n\nWe substitute $x = 8$ and $y = k$ into our equation:\n\n$$k + 5 = \\frac{2}{3}(8 - 2)$$\n\nThis is where you earn the first method mark (**M1**) for setting up a correct relationship to find $k$."},{"type":"text","order":3,"title":"Solve for k","content":"Now, let's simplify and solve for $k$:\n\n$$\\begin{aligned}\nk + 5 &= \\frac{2}{3}(6) \\\\\nk + 5 &= 4 \\quad \\text{(M1 for simplifying)} \\\\\nk &= 4 - 5 \\\\\nk &= -1 \\quad \\text{(A1 for final answer)}\n\\end{aligned}$$\n\nThus, the value of $k$ is $-1$."}]},{"id":"626bf0d8-3674-4a90-81b3-efd2536226be:rise_over_run","label":"Rise over Run Geometric Approach","steps":[{"type":"text","order":0,"title":"Find the horizontal run","content":"To find the horizontal run (the change in $x$) between the points $P(2, -5)$ and $Q(8, k)$, we subtract the $x$-coordinate of $P$ from the $x$-coordinate of $Q$:\n\n$$\\text{Run} = 8 - 2 = 6$$","highlights":[{"text":"P(2, -5)","location":"specification"},{"text":"Q(8, k)","location":"specification"}]},{"type":"text","order":1,"title":"Scale the gradient","content":"The gradient represents the ratio of vertical rise to horizontal run:\n\n$$\\text{Gradient} = \\frac{\\text{Rise}}{\\text{Run}} = \\frac{2}{3}$$\n\nWe know our actual horizontal run is $6$. We can set up an equivalent fraction to find the actual vertical rise:\n\n$$\\frac{2}{3} = \\frac{\\text{Rise}}{6}$$\n\nSince the denominator is multiplied by $2$ (from $3$ to $6$), we also multiply the numerator by $2$:\n\n$$\\text{Rise} = 2 \\times 2 = 4$$","highlights":[{"text":"The gradient of PQ is \\frac{2}{3}","location":"specification"}]},{"type":"text","order":2,"title":"Calculate the value of k","content":"Now, we add this vertical rise to the starting $y$-coordinate of point $P$, which is $-5$:\n\n$$\\begin{aligned} k &= y_P + \\text{Rise} \\\\ k &= -5 + 4 \\\\ k &= -1 \\end{aligned}$$\n\nThis matches the markscheme where:\n- Setting up the relationship earns M1\n- Finding the vertical change ($k + 5 = 4$) earns M1\n- Simplifying to the final answer $k = -1$ earns A1."}]}],"generatedAt":"2026-06-12T12:51:50.230Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":31}},{"id":"4d506551-6619-4c48-a3e6-e0314e524277","revisionId":"2d924d9b-0f4f-478b-8069-f3d677aef3b4","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"hard","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"2ab16267-44ea-431b-bfcd-c21443c7f367","label":"a","content":"Find the gradient of the line that passes through $(4, -3)$ and $(8, 9)$.","markscheme":"- Uses $m = \\frac{9 - (-3)}{8 - 4}$ M1\n- $\\frac{12}{4} = 3$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Evidence of u

Gradient of a Line

Question 2

Gradient of a Line Questions

Gradient of a Line

Question 2

Gradient of a Line Questions