tions":"1. Press [alpha] then [Y=] and select 1:n/d to open a fraction template.\n2. In the numerator, enter: 7 - 4\n3. Press the down arrow key.\n4. In the denominator, enter: -5 - (-3) (use the negative key [(-)] for the negative signs and the subtraction key [-] for minus).\n5. Press [ENTER]. To convert the fraction to a decimal, press [math], select 2:>Dec, and press [ENTER]."},"desmos":{"expression":"\\frac{7 - 4}{-5 - (-3)}","instructions":"Type `(7 - 4) / (-5 - (-3))` into the expression bar to see the result immediately."},"description":"You can perform this calculation easily on a calculator to avoid errors with signs."}]}]},{"id":"e8441c86-8e06-4b0d-9943-2c342e6a6401:simultaneous_equations","label":"Simultaneous Equations","steps":[{"type":"text","order":0,"title":"Set up general equation","content":"Any straight line can be written in the slope-intercept form:\n\n$$y = mx + c$$\n\nwhere $m$ is the gradient and $c$ is the $y$-intercept. \n\nWe can find $m$ by substituting the given points $(-3, 4)$ and $(-5, 7)$ into this general equation to create a system of two simultaneous equations."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\text{Using } (\\htmlData{anchor=x1}{\\color{@sky}{-3}}, \\htmlData{anchor=y1}{\\color{@pink}{4}}): \\quad \\htmlData{anchor=eq1}{\\color{@pink}{4} = \\color{@sky}{-3}m + c}"},{"latex":"\text{Using } (\\htmlData{anchor=x2}{\\color{@indigo}{-5}}, \\htmlData{anchor=y2}{\\color{@purple}{7}}): \\quad \\htmlData{anchor=eq2}{\\color{@purple}{7} = \\color{@indigo}{-5}m + c}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x1"},{"color":"pink","style":"text-color","anchor":"y1"},{"color":"indigo","style":"text-color","anchor":"x2"},{"color":"purple","style":"text-color","anchor":"y2"}],"connections":[]},"order":1,"title":"Form the equations","description":"Substitute $x = -3, y = 4$ for the first equation, and $x = -5, y = 7$ for the second equation."},{"type":"text","order":2,"title":"Solve for gradient","content":"To eliminate $c$, subtract the first equation from the second equation:\n\n$$\\begin{aligned}\n(7) - (4) &= (-5m + c) - (-3m + c) \\\\\n3 &= -5m + 3m \\\\\n3 &= -2m\n\\end{aligned}$$\n\nNow, solve for the gradient $m$:\n\n$$m = \\frac{3}{-2} = -1.5$$\n\nThis working earns you the first method mark M1 and the final accuracy mark A1."}]}],"generatedAt":"2026-06-12T21:11:17.582Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"725da9fd-00fd-42da-862f-85e078aff001","revisionId":"68887aac-8a53-40ae-9af5-f9e9046010af","questionNumber":"16","questionType":"LA","specification":"The line $AB$ passes through the points $A(2, -1)$ and $(6, k)$.\n\nThe gradient of $AB$ is 5.","difficulty":3,"difficultyBand":"hard","totalMarks":3,"sourceTitle":"Gradient of a Line","calculatorUse":"non_calc","parts":[{"id":"662b16f0-128a-47a2-ad37-f1d9d1f78479","label":"","content":"Work out the value of $k$.","markscheme":"- Sets up equation $5 = \\frac{k - (-1)}{6 - 2}$ M1\n- Simplifies to $5 = \\frac{k + 1}{4}$ and multiplies by 4 to get $20 = k + 1$ M1\n- $k = 19$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correct substitution of given values into the gradient formula.","code":"M"},{"id":"p2","marks":1,"description":"Correct algebraic step to isolate $k$.","code":"M"},{"id":"p3","marks":1,"description":"Final answer of 19.","code":"A"}],"finalAnswers":["19"],"generatedAt":"2026-05-06T11:06:51.378Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"gradient_formula","label":"Use the gradient formula","steps":[{"type":"text","order":0,"title":"Recall the formula","content":"For two points $(x_1,y_1)$ and $(x_2,y_2)$, use\n$$\\text{gradient}=\\frac{y_2-y_1}{x_2-x_1}.$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$5=\\dfrac{\\color{@orange}{\\htmlData{anchor=old_negative}{k-(-1)}}}{\\color{@green}{\\htmlData{anchor=old_run}{6-2}}}$"},{"latex":"$$5=\\dfrac{\\color{@orange}{\\ht

Gradient of a Line

Question 38

Gradient of a Line Questions

Gradient of a Line

Question 38

Gradient of a Line Questions