","color":"green","label":"divide by negative four"}]},"order":3,"title":"Find y","description":"Substitute the value of x into the simpler original equation to find the remaining coordinate."},{"type":"text","order":4,"title":"State the intersection","content":"The lines intersect at\n\n$$\\left(\\frac{7}{11},\\frac{9}{22}\\right).$$\n\nGiving the coordinates in this order earns the **A1** mark.","highlights":[{"text":"Find the coordinates of the point where $l_1$ and $l_2$ intersect.","location":"specification"}]}]}],"generatedAt":"2026-07-21T04:15:01.122Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":77}},{"id":"0ba66db8-5589-47be-8c40-8c22d1cfc796","revisionId":"8d56a695-bce9-4dc1-8137-018487f5e6cb","questionNumber":"8","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":5,"sourceTitle":"Question Set 5: Straight Line Graphs","calculatorUse":null,"parts":[{"id":"838b4639-883d-40d0-ac08-1724925d0635","label":"0","content":"The line $l_1$ has the equation $2x - 3y - 4 = 0$\nThe line $l_2$ is perpendicular to $l_1$ and passes through the point $(4, -1)$\n\nFind an equation for $l_2$ in the form $ax + by + c = 0$","markscheme":"- Gradient of $l_1$: $3y = 2x - 4 \\Rightarrow m = \\frac{2}{3}$ M1\n- Perpendicular gradient $= -\\frac{3}{2}$ M1\n- $y + 1 = -\\frac{3}{2}(x - 4)$ M1\n- $2y + 2 = -3x + 12$ M1\n- $3x + 2y - 10 = 0$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Finds the gradient of $l_1$.","code":"M"},{"id":"p2","marks":1,"description":"Determines the perpendicular gradient.","code":"M"},{"id":"p3","marks":1,"description":"Uses the point $(4, -1)$ and perpendicular gradient to form an equation.","code":"M"},{"id":"p4","marks":1,"description":"Clears fractions from the equation.","code":"M"},{"id":"p5","marks":1,"description":"Provides the final equation $3x + 2y - 10 = 0$.","code":"A"}],"finalAnswers":["3x + 2y - 10 = 0"],"generatedAt":"2026-05-12T02:14:06.602Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"perpendicular_line","label":"Use perpendicular gradients","steps":[{"type":"text","order":0,"title":"Rearrange the line","content":"Start with the equation of $l_1$ and make $y$ the subject:\n\n$$2x-3y-4=0$$\n\n$$3y=2x-4$$\n\n$$y=\\frac{2}{3}x-\\frac{4}{3}$$\n\nSo the gradient of $l_1$ is $\\frac{2}{3}$. Writing this rearrangement earns the first method mark.","highlights":[{"text":"The line $l_1$ has the equation $2x - 3y - 4 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"m_1=\\frac{2}{3}"},{"latex":"m_2=-\\frac{1}{m_1}"},{"latex":"m_2=\\htmlData{anchor=perpendicular_slope}{\\color{@green}{-\\frac{3}{2}}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"perpendicular_slope"}]},"order":1,"title":"Use perpendicular gradients","description":"Perpendicular non-vertical lines have gradients whose product is negative one. Therefore, take the negative reciprocal of two-thirds; this earns the second method mark."},{"type":"text","order":2,"title":"Use the given point","content":"The new line passes through $(4,-1)$ and has gradient $-\\frac{3}{2}$. Use the point-slope form $y-y_1=m(x-x_1)$:\n\n$$y-(-1)=-\\frac{3}{2}(x-4)$$\n\nwhich is\n\n$$y+1=-\\frac{3}{2}(x-4).$$\n\nWriting this substitution earns the third method mark.","highlights":[{"text":"passes through the point $(4, -1)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"y+1=-\\frac{3}{2}(x-4)"},{"latex":"2(y+1)=-3(x-4)"}]},"order":3,"title":"Clear the denominator","description":"Multiply both sides by 2 to clear the denominator. This keeps the equation equivalent."},{"type":"hyper_latex","block":{"nodes":[{"latex":"2(y+1)=-3(x-4)"},{"latex":"2y+2=-3x+12"}]},"order":4,"title":"Expand both brackets","description":"Expand both brackets to obtain the next line. This earns

Straight Line Graphs

Question 5

Straight Line Graphs Questions

Straight Line Graphs

Question 5

Straight Line Graphs Questions