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A larger gradient magnitude means the straight line is steeper."},{"text":"The height of the line (the value of $y$ where the line crosses the $y$-axis).","correct":false,"explanation":"The value where a line crosses the $y$-axis is its $y$-intercept, not its gradient. Gradient measures steepness through the change in $y$ per unit change in $x$."},{"text":"The length of the line (the distance between the two endpoints of the line segment).","correct":false,"explanation":"Length measures the distance between two points and depends on both coordinate changes. Gradient instead measures steepness by comparing the vertical change with the horizontal change."},{"text":"The position of the line (how far the line is from the origin of the graph).","correct":false,"explanation":"A line’s position relative to the origin does not determine its gradient. Parallel lines can occupy different positions while sharing the same steepness and gradient."}]},{"id":"a27c3b4c-20b0-4576-b19f-6714894d78a4","cardType":"basic","points":1,"front":"State the simple formula for gradient using vertical and horizontal change.","back":"$$$\\text{gradient} = \\frac{\\text{rise}}{\\text{run}}$$","mcqOptions":[{"text":"$$$\\text{gradient} = \\frac{\\text{rise}}{\\text{run}}$$","correct":true,"explanation":"The rise is the vertical change and the run is the horizontal change. Dividing rise by run gives the change in $y$ for each unit change in $x$."},{"text":"$$$\\text{gradient} = \\frac{\\text{run}}{\\text{rise}}$$","correct":false,"explanation":"This reverses the numerator and denominator, producing the reciprocal of the gradient. The correct formula divides the vertical change, or rise, by the horizontal change, or run."},{"text":"$$$\\text{gradient} = \\text{rise} \\times \\text{run}$$","correct":false,"explanation":"Multiplying the coordinate changes does not measure the line’s rate of change. Gradient is a ratio, so the rise must be divided by the run."},{"text":"$$$\\text{gradient} = \\text{rise} + \\text{run}$$","correct":false,"explanation":"Adding the changes does not compare vertical movement with horizontal movement. The gradient is calculated using the ratio $\\frac{\\text{rise}}{\\text{run}}$."}]},{"id":"f9aa368d-d6c1-4b8c-b3ef-713049a8654e","cardType":"basic","points":1,"front":"When finding the gradient from a graph, how should you choose your two points?","back":"Pick points where the line passes exactly through grid intersections.","mcqOptions":[{"text":"Pick points where the line passes exactly through grid intersections.","correct":true,"explanation":"Grid intersections give coordinates that can be read accurately. Using two such points on the line reduces estimation errors when calculating rise and run."},{"text":"Pick points that are close to the line at grid intersections.","correct":false,"explanation":"Nearby grid intersections cannot be used unless the line actually passes through them. The correct method is to choose two grid intersections lying exactly on the line."},{"text":"Pick points where the line is nearest to the coordinate axes.","correct":false,"explanation":"Distance from the coordinate axes does not make a point suitable. Choose points where the line passes exactly through grid intersections so their coordinates are precise."},{"text":"Pick points that lie halfway between two grid intersections.","correct":false,"explanation":"Halfway positions are harder to read accurately and can introduce estimation errors. Points where the line passes exactly through grid intersections should be chosen instead."}]},{"id":"3861dbff-9ae1-4851-aaf4-066fc73f1273","cardType":"cloze","points":null,"front":"In the formula $\\text{gradient} = \\frac{\\text{rise}}{\\text{run}}$, the rise is the [ ] change and the run is the horizontal change.","back":"In the formula $\\text{gradient} = \\frac{\\text{rise}}{\\text{run}}$, the rise is the vertical change and the run is the horizontal change.","mcqOptions":[{"text":"vertical","correct":true,"explanation":"Rise measures the change in the $y$-direction, so it is the vertical change between the two selected points on the line."},{"text":"horizontal","correct":false,"explanation":"Horizontal change is the run, not the rise. Rise measures how far the line moves vertically between the two chosen points."},{"text":"diagonal","correct":false,"explanation":"The diagonal distance between the points is not used directly in the gradient formula. Rise is specifically the vertical change in the $y$-direction."},{"text":"total","correct":false,"explanation":"Gradient does not use a total of the coordinate changes. Rise refers only to the vertical change between the selected points."}]},{"id":"0021e8f8-eaa2-42b0-966f-2bbbc7da4f5e","cardType":"cloze","points":null,"front":"In the formula $\\text{gradient} = \\frac{\\text{rise}}{\\text{run}}$, the rise is the vertical change and the run is the [ ] change.","back":"In the formula $\\text{gradient} = \\frac{\\text{rise}}{\\text{run}}$, the rise is the vertical change and the run is the horizontal change.","mcqOptions":[{"text":"horizontal","correct":true,"explanation":"Run measures the change in the $x$-direction, so it is the horizontal change between the two selected points on the line."},{"text":"vertical","correct":false,"explanation":"Vertical change is the rise, not the run. Run measures how far the line moves horizontally between the two chosen points."},{"text":"diagonal","correct":false,"explanation":"The diagonal distance between points is not the run. Run is specifically the horizontal change in the $x$-direction."},{"text":"total","correct":false,"explanation":"Run is not the total distance or the sum of coordinate changes. It refers only to the horizontal change between the selected points."}]},{"id":"41f8807a-48d6-4802-9f4e-308bbc80c4cb","cardType":"basic","points":1,"front":"How can you tell if a line has a positive gradient just by looking at a graph?","back":"The line slopes upwards from left to right.","mcqOptions":[{"text":"The line slopes upwards from left to right.","correct":true,"explanation":"As $x$ increases from left to right, $y$ also increases. Therefore the rise and run have the same sign, giving a positive gradient."},{"text":"The line slopes downwards from left to right.","correct":false,"explanation":"When a line slopes downwards from left to right, $y$ decreases as $x$ increases, so its gradient is negative. A positive-gradient line slopes upwards."},{"text":"The line stays horizontal from left to right.","correct":false,"explanation":"A horizontal line has no vertical change, so its rise and gradient are both zero. A positive-gradient line must slope upwards from left to right."},{"text":"The line slopes upwards from right to left.","correct":false,"explanation":"Sloping upwards from right to left means the line slopes downwards when viewed from left to right, giving a negative gradient rather than a positive one."}]},{"id":"dce2404a-0a87-4f1c-a9a8-aa126c63e1a3","cardType":"basic","points":1,"front":"How can you tell if a line has a negative gradient just by looking at a graph?","back":"The line slopes downwards from left to right.","mcqOptions":[{"text":"The line slopes downwards from left to right.","correct":true,"explanation":"As $x$ increases from left to right, $y$ decreases. A positive run combined with a negative rise gives a negative gradient."},{"text":"The line slopes upwards from left to right.","correct":false,"explanation":"When a line slopes upwards from left to right, $y$ increases as $x$ increases, giving a positive gradient. A negative-gradient line slopes downwards."},{"text":"The line stays horizontal from left to right.","correct":false,"explanation":"A horizontal line has a rise of zero, so its gradient is zero rather than negative. A negative-gradient line slopes downwards from left to right."},{"text":"The line slopes downwards from right to left.","correct":false,"explanation":"Sloping downwards from right to left means the line slopes upwards when viewed from left to right, so its gradient is positive rather than negative."}]},{"id":"f16bbc6b-cfff-421b-828e-a9b65dc6a0bd","cardType":"basic","points":2,"front":"What is the common mistake when calculating the rise of a line that slopes downwards?","back":"Counting the vertical change as positive instead of negative.","mcqOptions":[{"text":"Counting the vertical change as positive instead of negative.","correct":true,"explanation":"Moving from left to right along a downward-sloping line makes $y$ decrease, so the rise is negative. Treating it as positive incorrectly produces a positive gradient."},{"text":"Counting the vertical change as negative instead of positive.","correct":false,"explanation":"For a downward-sloping line measured from left to right, the vertical change should be negative. The common error is wrongly recording this fall as a positive rise."},{"text":"Counting the vertical change as zero instead of negative.","correct":false,"explanation":"A zero vertical change applies only to a horizontal line. The usual mistake for a downward-sloping line is treating the negative vertical change as positive."},{"text":"Counting the horizontal change as negative instead of positive.","correct":false,"explanation":"This concerns the run rather than the rise. When working from left to right, the run is positive, while the common rise error is ignoring its negative sign."}]},{"id":"a6f18846-9d8c-4eba-8fc7-0d5506ed86ea","cardType":"cloze","points":null,"front":"If a line goes up 1 square for every 3 squares it moves to the right, its gradient is $\\text{[...]}$.","back":"If a line goes up 1 square for every 3 squares it moves to the right, its gradient is $\\boxed{\\frac{1}{3}}$.","mcqOptions":[{"text":"$$\\frac{1}{3}$","correct":true,"explanation":"Gradient is rise divided by run. The line rises by 1 square while moving 3 squares right, so its gradient is $\\frac{1}{3}$."},{"text":"$$-\\frac{1}{3}$","correct":false,"explanation":"A negative gradient would mean the line falls as it moves from left to right. This line rises, so rise over run gives $\\frac{1}{3}$."},{"text":"$$3$","correct":false,"explanation":"This reverses rise and run by calculating run divided by rise. Gradient is rise divided by run, giving $\\frac{1}{3}$ rather than $3$."},{"text":"$$-3$","correct":false,"explanation":"This both reverses rise and run and assigns the wrong sign. The line rises to the right, so its gradient is positive and equals $\\frac{1}{3}$."}]},{"id":"cf97f65e-014e-4336-b018-e6be332a4142","cardType":"basic","points":1,"front":"State the coordinate formula for gradient between $(x_1, y_1)$ and $(x_2, y_2)$.","back":"$$$\\text{gradient} = \\frac{y_2 - y_1}{x_2 - x_1}$$","mcqOptions":[{"text":"$$$\\text{gradient} = \\frac{y_2 - y_1}{x_2 - x_1}$$","correct":true,"explanation":"Gradient is the change in the $y$-coordinates divided by the corresponding change in the $x$-coordinates, with the points subtracted in the same order."},{"text":"$$$\\text{gradient} = \\frac{x_2 - x_1}{y_2 - y_1}$$","correct":false,"explanation":"This reverses the fraction and calculates run divided by rise. Gradient must be change in $y$ divided by change in $x$: $\\frac{y_2-y_1}{x_2-x_1}$."},{"text":"$$$\\text{gradient} = \\frac{y_2 + y_1}{x_2 + x_1}$$","correct":false,"explanation":"Adding coordinates does not measure the change between two points. Subtract corresponding coordinates in a consistent order to obtain $\\frac{y_2-y_1}{x_2-x_1}$."},{"text":"$$$\\text{gradient} = \\frac{y_2 - y_1}{x_1 - x_2}$$","correct":false,"explanation":"The numerator and denominator use opposite subtraction orders here, incorrectly changing the sign. Both differences must follow the same point order: $\\frac{y_2-y_1}{x_2-x_1}$."}]},{"id":"bb40d29c-4cfe-49ef-8dc3-f630db262cc9","cardType":"basic","points":2,"front":"What is the consistency rule when using the coordinate formula $\\frac{y_2 - y_1}{x_2 - x_1}$?","back":"The order of points must be the same for both the top and bottom (e.g. if you start with $y_2$, you must start with $x_2$).","mcqOptions":[{"text":"The order of points must be the same for both the top and bottom (e.g. if you start with $y_2$, you must start with $x_2$).","correct":true,"explanation":"Using the same point order preserves the correct sign because both coordinate differences describe movement in one consistent direction between the two points."},{"text":"The smaller coordinate must come first for both the top and bottom (e.g. if $y_1$ is smaller, you must start with $x_1$).","correct":false,"explanation":"Starting with the smaller coordinate is unnecessary and may produce inconsistent subtraction orders. The same point must come first in both the $y$-difference and the $x$-difference."},{"text":"The second point must always come first for both the top and bottom (e.g. you must start with $y_2$ and then start with $x_2$).","correct":false,"explanation":"The second point does not always have to come first; either point may be chosen first. What matters is using that same order in both coordinate differences."},{"text":"The order of points must be opposite for the top and bottom (e.g. if you start with $y_2$, you must start with $x_1$).","correct":false,"explanation":"Opposite subtraction orders introduce an incorrect negative sign. If the numerator starts with $y_2$, the denominator must start with $x_2$, not $x_1$."}]},{"id":"4d1a4b22-f74c-4246-8e12-7e9300b12309","cardType":"basic","points":1,"front":"What is the gradient of any horizontal line?","back":"0","mcqOptions":[{"text":"0","correct":true,"explanation":"A horizontal line has no vertical change, so its rise is zero. Dividing zero by any non-zero horizontal run gives a gradient of 0."},{"text":"1","correct":false,"explanation":"A gradient of 1 means the line rises one unit for every unit moved right. A horizontal line has zero rise, so its gradient is 0."},{"text":"Undefined","correct":false,"explanation":"Undefined gradient applies to a vertical line because its horizontal change is zero. A horizontal line has zero vertical change, so its gradient is 0."},{"text":"-1","correct":false,"explanation":"A gradient of -1 describes a line falling one unit for every unit moved right. A horizontal line neither rises nor falls, giving a gradient of 0."}]},{"id":"6849a215-9a29-4b05-a27b-b5acbedede34","cardType":"basic","points":2,"front":"Why is the gradient of a vertical line considered undefined/not allowed?","back":"The change in $x$ is 0, and you cannot divide by zero.","mcqOptions":[{"text":"The change in $x$ is 0, and you cannot divide by zero.","correct":true,"explanation":"Every pair of points on a vertical line has the same $x$-coordinate, making the run zero. The gradient calculation would therefore require division by zero."},{"text":"The change in $y$ is 0, and you cannot divide by zero.","correct":false,"explanation":"Zero change in $y$ describes a horizontal line, whose gradient is 0. A vertical line instead has zero change in $x$, causing division by zero."},{"text":"The changes in $x$ and $y$ are equal, and the gradient must be 1.","correct":false,"explanation":"Equal changes in $x$ and $y$ would produce a gradient of 1, not an undefined gradient. A vertical line has zero change in $x$."},{"text":"The line has no $y$-intercept, and every line must cross the $y$-axis.","correct":false,"explanation":"A line does not need a $y$-intercept for its gradient to exist. Vertical gradients are undefined specifically because the change in $x$ is zero."}]},{"id":"25df227c-de23-46b9-93c9-d6da7f8432eb","cardType":"cloze","points":null,"front":"A horizontal line has a gradient of [ ] because its $y$-value does not change.","back":"A horizontal line has a gradient of 0 because its $y$-value does not change.","mcqOptions":[{"text":"0","correct":true,"explanation":"A horizontal line’s $y$-value remains constant, so the rise is zero. Gradient is rise divided by run, and zero divided by a non-zero run is 0."},{"text":"1","correct":false,"explanation":"A gradient of 1 requires equal vertical and horizontal changes. A horizontal line has no vertical change, so its gradient is 0 rather than 1."},{"text":"undefined","correct":false,"explanation":"Undefined gradient is associated with a vertical line, where the run is zero. A horizontal line has zero rise and therefore has a gradient of 0."},{"text":"-1","correct":false,"explanation":"A gradient of -1 means the line falls as it moves right. A horizontal line has no change in height, so its gradient is 0."}]},{"id":"69f42b30-5e40-4ce0-80ea-a19a99f0ed1e","cardType":"cloze","points":null,"front":"A horizontal line has a gradient of 0 because its [ ] does not change.","back":"A horizontal line has a gradient of 0 because its $y$-value does not change.","mcqOptions":[{"text":"$$y$-value","correct":true,"explanation":"All points on a horizontal line have the same $y$-coordinate. Therefore, the vertical change is zero and the gradient is 0."},{"text":"$$x$-value","correct":false,"explanation":"The $x$-value changes as points move along a horizontal line. It is the $y$-value that remains constant, making the gradient 0."},{"text":"$$x$-intercept","correct":false,"explanation":"The $x$-intercept is a fixed point rather than the coordinate whose constancy defines a horizontal line. Its $y$-value remains unchanged."},{"text":"run","correct":false,"explanation":"The run can vary depending on which two points are selected. The rise is always zero because the $y$-value remains constant."}]},{"id":"08f3927d-29a4-42fb-920d-cdf1e3bd5841","cardType":"basic","points":2,"front":"Calculate the gradient of the line passing through $(-2, -3)$ and $(0, 5)$.","back":"4 (Rise is $5 - (-3) = 8$; Run is $0 - (-2) = 2$; $\\frac{8}{2} = 4$)","mcqOptions":[{"text":"4 (Rise is $5 - (-3) = 8$; Run is $0 - (-2) = 2$; $\\frac{8}{2} = 4$)","correct":true,"explanation":"The rise is $5-(-3)=8$ and the run is $0-(-2)=2$. Dividing rise by run gives $\\frac{8}{2}=4$."},{"text":"-4 (Rise is $-3 - 5 = -8$; Run is $0 - (-2) = 2$; $\\frac{-8}{2} = -4$)","correct":false,"explanation":"This sign error comes from reversing only the $y$-subtraction while leaving the $x$-subtraction unchanged. Both differences need the same order, giving a positive gradient of 4."},{"text":"2 (Rise is $5 - 3 = 2$; Run is $0 - (-2) = 2$; $\\frac{2}{2} = 2$)","correct":false,"explanation":"This treats subtracting the negative coordinate as ordinary subtraction. Since $5-(-3)=5+3=8$, the rise is 8 and the gradient is 4."},{"text":"$$\\frac{1}{4}$ (Rise is $5 - (-3) = 8$; Run is $0 - (-2) = 2$; $\\frac{2}{8} = \\frac{1}{4}$)","correct":false,"explanation":"This calculates run divided by rise, reversing the gradient formula. Gradient is rise divided by run, so the correct calculation is $\\frac{8}{2}=4$."}]},{"id":"2c7df678-dd89-4d0f-a448-db3c68a74f6c","cardType":"basic","points":2,"front":"Calculate the gradient of the line passing through $(-4, 7)$ and $(2, -5)$.","back":"-2 (Rise is $-5 - 7 = -12$; Run is $2 - (-4) = 6$; $\\frac{-12}{6} = -2$)","mcqOptions":[{"text":"-2 (Rise is $-5 - 7 = -12$; Run is $2 - (-4) = 6$; $\\frac{-12}{6} = -2$)","correct":true,"explanation":"Gradient is change in $y$ divided by change in $x$. The rise is $-12$ and the run is $6$, giving $\\frac{-12}{6}=-2$."},{"text":"2 (Rise is $7 - (-5) = 12$; Run is $2 - (-4) = 6$; $\\frac{12}{6} = 2$)","correct":false,"explanation":"The $y$-coordinates were subtracted in the opposite order from the $x$-coordinates, incorrectly changing the sign. Consistent subtraction gives $\\frac{-12}{6}=-2$."},{"text":"$$-\\frac{1}{2}$ (Rise is $2 - (-4) = 6$; Run is $-5 - 7 = -12$; $\\frac{6}{-12} = -\\frac{1}{2}$)","correct":false,"explanation":"This reverses rise and run, calculating change in $x$ divided by change in $y$. Gradient is $\\frac{-12}{6}=-2$, not its reciprocal."},{"text":"-3 (Rise is $-5 - 7 = -12$; Run is $2 - (-4) = 4$; $\\frac{-12}{4} = -3$)","correct":false,"explanation":"The run was calculated without recognising that subtracting $-4$ means adding $4$. The run is $6$, so the gradient is $\\frac{-12}{6}=-2$."}]},{"id":"8cdf64e2-692c-40d3-a6e0-d73c7fc68993","cardType":"cloze","points":null,"front":"The gradient measures how much the [ ] changes for each increase of 1 in the $x$-value.","back":"The gradient measures how much the $y$-value changes for each increase of 1 in the $x$-value.","mcqOptions":[{"text":"$$y$-value","correct":true,"explanation":"Gradient describes the vertical change per unit of horizontal change, so it measures how much the $y$-value changes whenever the $x$-value increases by 1."},{"text":"$$x$-value","correct":false,"explanation":"The $x$-value provides the horizontal increase rather than the quantity being measured. Gradient measures the change in the $y$-value for each unit increase in $x$."},{"text":"$$y$-intercept","correct":false,"explanation":"The $y$-intercept is the fixed point where a line crosses the $y$-axis. Gradient instead measures how much the $y$-value changes per unit increase in $x$."},{"text":"$$x$-intercept","correct":false,"explanation":"The $x$-intercept identifies where the line crosses the $x$-axis and does not describe its steepness. Gradient measures the change in the $y$-value per unit increase in $x$."}]},{"id":"3b74b719-c7d6-4412-85b8-7247d87b5d7c","cardType":"cloze","points":null,"front":"The gradient measures how much the $y$-value changes for each [ ] in the $x$-value.","back":"The gradient measures how much the $y$-value changes for each increase of 1 in the $x$-value.","mcqOptions":[{"text":"increase of 1","correct":true,"explanation":"Gradient is the change in $y$ for one unit of positive horizontal movement. It therefore measures how much the $y$-value changes for each increase of 1 in $x$."},{"text":"decrease of 1","correct":false,"explanation":"Using a decrease of 1 reverses the direction of horizontal movement and would reverse the sign of the measured change. Gradient is defined here for an increase of 1 in $x$."},{"text":"increase of 2","correct":false,"explanation":"An increase of 2 would produce twice the vertical change for a straight line, not the gradient itself. Gradient gives the change in $y$ for an increase of 1 in $x$."},{"text":"decrease of 2","correct":false,"explanation":"This changes both the direction and the size of the horizontal interval. Gradient measures vertical change for a positive increase of exactly 1 in the $x$-value."}]},{"id":"630de689-f4b3-422e-b6a8-6320db2016c7","cardType":"basic","points":3,"front":"The line through $A(3, -2)$ and $B(7, k)$ has gradient 4. Outline the steps to find $k$.","back":"

Set up the formula: $4 = \\frac{k - (-2)}{7 - 3}$
Simplify: $4 = \\frac{k + 2}{4}$
Solve: $16 = k + 2$, so $k = 14$

","mcqOptions":[{"text":"

Set up the formula: $4 = \\frac{k - (-2)}{7 - 3}$
Simplify: $4 = \\frac{k + 2}{4}$
Solve: $16 = k + 2$, so $k = 14$

","correct":true,"explanation":"Substituting both points into $m=\\frac{y_2-y_1}{x_2-x_1}$ gives $4=\\frac{k+2}{4}$. Multiplying by 4 and subtracting 2 gives $k=14$."},{"text":"

Set up the formula: $4 = \\frac{k - 2}{7 - 3}$
Simplify: $4 = \\frac{k - 2}{4}$
Solve: $16 = k - 2$, so $k = 18$

","correct":false,"explanation":"The numerator contains $k-(-2)$, not $k-2$; subtracting a negative gives $k+2$. Therefore $16=k+2$, so the correct value is $k=14$."},{"text":"

Set up the formula: $4 = \\frac{k - (-2)}{3 - 7}$
Simplify: $4 = \\frac{k + 2}{-4}$
Solve: $-16 = k + 2$, so $k = -18$

","correct":false,"explanation":"The $x$-coordinates were reversed while the $y$-coordinates were not, incorrectly changing the gradient’s sign. Using the same subtraction order gives $4=\\frac{k+2}{4}$ and $k=14$."},{"text":"

Set up the formula: $4 = \\frac{k - (-2)}{7 + 3}$
Simplify: $4 = \\frac{k + 2}{10}$
Solve: $40 = k + 2$, so $k = 38$

","correct":false,"explanation":"The run is the difference $7-3=4$, not the sum $7+3=10$. Thus $4=\\frac{k+2}{4}$, which leads to the correct value $k=14$."}]},{"id":"74fb96f9-0f2b-4d22-91e0-cc27ce8ce998","cardType":"basic","points":3,"front":"The line through $(-1, 6)$ and $(k, 18)$ has gradient 3. Outline the steps to find $k$.","back":"

Set up the formula: $3 = \\frac{18 - 6}{k - (-1)}$
Simplify: $3 = \\frac{12}{k + 1}$
Solve: $3(k + 1) = 12 \\rightarrow k + 1 = 4$, so $k = 3$

","mcqOptions":[{"text":"

Set up the formula: $3 = \\frac{18 - 6}{k - (-1)}$
Simplify: $3 = \\frac{12}{k + 1}$
Solve: $3(k + 1) = 12 \\rightarrow k + 1 = 4$, so $k = 3$

","correct":true,"explanation":"The vertical change is $12$ and the horizontal change is $k+1$. Solving $3=\\frac{12}{k+1}$ gives $k+1=4$, so $k=3$."},{"text":"

Set up the formula: $3 = \\frac{18 - 6}{k - 1}$
Simplify: $3 = \\frac{12}{k - 1}$
Solve: $3(k - 1) = 12 \\rightarrow k - 1 = 4$, so $k = 5$

","correct":false,"explanation":"The horizontal change is $k-(-1)=k+1$, because subtracting a negative becomes addition. Solving $3=\\frac{12}{k+1}$ gives the correct value $k=3$."},{"text":"

Set up the formula: $3 = \\frac{18 + 6}{k - (-1)}$
Simplify: $3 = \\frac{24}{k + 1}$
Solve: $3(k + 1) = 24 \\rightarrow k + 1 = 8$, so $k = 7$

","correct":false,"explanation":"The rise is the difference between the $y$-coordinates, $18-6=12$, not their sum. Using $12$ gives $k+1=4$, so the correct value is $k=3$."},{"text":"

Set up the formula: $3 = \\frac{k - (-1)}{18 - 6}$
Simplify: $3 = \\frac{k + 1}{12}$
Solve: $36 = k + 1 \\rightarrow k + 1 = 36$, so $k = 35$

","correct":false,"explanation":"This reverses the gradient formula by dividing the run by the rise. Gradient is rise over run, so $3=\\frac{12}{k+1}$ and the correct value is $k=3$."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Gradient of a Line","title":"Gradient of a Line Flashcards","topbar":{"backHref":"/igcse/maths/edexcel/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"49af1960-7f5b-407b-8bc0-7c9b35604868","walled":"$L39"}],"$L3a","$L3b"]