GCSE Maths: Gradient (Higher)
Part of the GCSE Maths straight-line graphs and coordinates chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Gradient
The gradient of a line tells you how steep it is. A steep hill has a big gradient; a gentle slope has a small one; flat ground has a gradient of zero. To measure it, compare how far the line goes up with how far it goes across:
gradient = rise ÷ run = (change in y) ÷ (change in x)
The rise is the vertical change and the run is the horizontal change. Always move from left to right, so the run is positive. Then the rise tells you the direction: if the line goes up the rise is positive, and if it goes down the rise is negative.
Finding the gradient from a graph
Choose two points on the line that sit exactly on grid corners, as far apart as you sensibly can. Draw a right-angled triangle under the line, count the squares for the run and the rise, and divide. The picture shows a line with a rise of 6 for a run of 3, so its gradient is 6 ÷ 3 = 2: it goes up 2 for every 1 step across.
Finding the gradient from two points
You do not need a picture. If the points are (x1, y1) and (x2, y2), the rise is y2 − y1 and the run is x2 − x1:
gradient = (y2 − y1) ÷ (x2 − x1)
It does not matter which point you call point 1, as long as you subtract in the same order on the top and on the bottom. Simplify the fraction at the end.
Special cases
A horizontal line has a rise of 0, so its gradient is 0. A vertical line has a run of 0, and you cannot divide by 0, so a vertical line has no gradient (we say it is undefined). Gradients are often written as fractions, for example 2/3 or −3/4. Road signs show the gradient as a percentage: a gradient of 0.08 is “8%”.
Key ideas
- Gradient = rise ÷ run = (change in y) ÷ (change in x). Read the line left to right.
- Up from left to right: positive. Down from left to right: negative. Horizontal: 0. Vertical: undefined.
- From two points: (y2 − y1) ÷ (x2 − x1), subtracting in the same order top and bottom.
- The steeper the line, the bigger the gradient (ignoring the sign).
- Simplify gradient fractions, and give the sign.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run. Draw a triangle on the line and count the squares up (or down) and across.
- Choose two points on the line that are exactly on grid corners. Points that sit on crossing grid lines are easy to read accurately. Choose them far apart for the best accuracy.(−1, −3) and (3, 5)
- Draw a right-angled triangle under the line and count the run. The run is the distance across, moving to the right.run = 3 − (−1) = 4
- Count the rise. The rise is the distance up (positive) or down (negative) while moving right along the line.rise = 5 − (−3) = 8
- Mark the triangle on the graph.
- Divide the rise by the run. Gradient = rise ÷ run, simplified.gradient = 8 ÷ 4 = 2
Check. The line goes up from left to right, so the gradient should be positive. Also, for every 1 step across the line goes up 2. ✓
Watch out. If the line goes down as you move right, the rise is negative. Always read the line from left to right.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run. Draw a triangle on the line and count the squares up (or down) and across.
- Choose two points on the line that are exactly on grid corners. Points that sit on crossing grid lines are easy to read accurately. Choose them far apart for the best accuracy.(−2, 7) and (4, −5)
- Draw a right-angled triangle under the line and count the run. The run is the distance across, moving to the right.run = 4 − (−2) = 6
- Count the rise. The rise is the distance up (positive) or down (negative) while moving right along the line.rise = −5 − 7 = -12
- Mark the triangle on the graph.
- Divide the rise by the run. Gradient = rise ÷ run, simplified.gradient = -12 ÷ 6 = −2
Check. The line goes down from left to right, so the gradient should be negative. Also, for every 1 step across the line goes down 2. ✓
Watch out. If the line goes down as you move right, the rise is negative. Always read the line from left to right.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run = (change in y) ÷ (change in x) = (y2 − y1) ÷ (x2 − x1).
- Label the points. Call the first point (A) (x1, y1) and the second (B) (x2, y2). It does not matter which is first, but you must be consistent.A = (−3, −4), B = (5, 0)
- Find the rise: the change in y. Rise is how far the line goes up (positive) or down (negative). Subtract the y-coordinates, second minus first.rise = y2 − y1 = 0 − (−4) = 4
- Find the run: the change in x. Run is how far across to the right. Subtract the x-coordinates in the same order as the y-coordinates.run = x2 − x1 = 5 − (−3) = 8
- Sketch the rise and run. The dashed triangle shows the run (across) and the rise (up or down) between the two points.
- Divide the rise by the run and simplify. Gradient = rise ÷ run. Cancel the fraction if you can. A negative divided by a positive (or the reverse) is negative.gradient = rise ÷ run = 4 ÷ 8 = 12
Check. Swap the points: (−4 − 0) ÷ (−3 − 5) = −4 ÷ (−8) = 12, the same. The gradient is positive: the line goes up from left to right. ✓
Watch out. Subtract in the same order on the top and the bottom. Mixing the order (y2 − y1 over x1 − x2) gives the wrong sign.
Method. Gradient = rise ÷ run. Both must be in the same units, so convert the rise to metres first.
- Write the rise and the run in the same units. A fraction compares the vertical change with the horizontal change. Here both are in metres.rise = 15 m, run = 200 m
- Write the gradient as a fraction. Gradient = rise ÷ run.15 ÷ 200 = 15200
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 5.15200 = 340
- Convert to a percentage if needed. A road gradient is often given as a percentage: multiply the decimal by 100.3 ÷ 40 = 0.075, so 0.075 × 100 = 7.5%
Check. Going 200 m along the flat, the road rises 7.5 m for every 100 m, which is 15 m in 200 m. ✓
Watch out. Rise and run must use the same units: do not divide metres by kilometres.
Watch out
Rise over run, never run over rise. The gradient is (up) ÷ (across). Also take care with signs: a line that goes down as you move right has a negative gradient. And always subtract in the same order on the top and the bottom of the fraction: y2 − y1 goes with x2 − x1.Exercise 12E Gradient
Non-calculatorKey words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Read the line from left to right: up is positive, down is negative, flat is zero, and vertical has no gradient.
- Look at line A from left to right. Read the line the way you read a sentence: from the left to the right.it goes up
- Decide the gradient. Going up as you move right means a positive gradient.The gradient of line A is positive.
Check. Imagine walking along the line from left to right: uphill, downhill or flat. ✓
Watch out. Always read from left to right. Reading right to left reverses the sign.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Read the line from left to right: up is positive, down is negative, flat is zero, and vertical has no gradient.
- Look at line B from left to right. Read the line the way you read a sentence: from the left to the right.it goes down
- Decide the gradient. Going down as you move right means a negative gradient.The gradient of line B is negative.
Check. Imagine walking along the line from left to right: uphill, downhill or flat. ✓
Watch out. Always read from left to right. Reading right to left reverses the sign.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Read the line from left to right: up is positive, down is negative, flat is zero, and vertical has no gradient.
- Look at line C from left to right. Read the line the way you read a sentence: from the left to the right.it stays level
- Decide the gradient. A horizontal line has a rise of 0, so the gradient is 0.The gradient of line C is zero.
Check. Imagine walking along the line from left to right: uphill, downhill or flat. ✓
Watch out. Always read from left to right. Reading right to left reverses the sign.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Read the line from left to right: up is positive, down is negative, flat is zero, and vertical has no gradient.
- Look at line D from left to right. Read the line the way you read a sentence: from the left to the right.it goes straight up and down
- Decide the gradient. A vertical line has a run of 0. We cannot divide by 0, so the gradient is undefined.The gradient of line D is undefined.
Check. Imagine walking along the line from left to right: uphill, downhill or flat. ✓
Watch out. Always read from left to right. Reading right to left reverses the sign.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run. Draw a triangle on the line and count the squares up (or down) and across.
- Choose two points on the line that are exactly on grid corners. Points that sit on crossing grid lines are easy to read accurately. Choose them far apart for the best accuracy.(−3, −5) and (2, 5)
- Draw a right-angled triangle under the line and count the run. The run is the distance across, moving to the right.run = 2 − (−3) = 5
- Count the rise. The rise is the distance up (positive) or down (negative) while moving right along the line.rise = 5 − (−5) = 10
- Mark the triangle on the graph.
- Divide the rise by the run. Gradient = rise ÷ run, simplified.gradient = 10 ÷ 5 = 2
Check. The line goes up from left to right, so the gradient should be positive. Also, for every 1 step across the line goes up 2. ✓
Watch out. If the line goes down as you move right, the rise is negative. Always read the line from left to right.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run. Draw a triangle on the line and count the squares up (or down) and across.
- Choose two points on the line that are exactly on grid corners. Points that sit on crossing grid lines are easy to read accurately. Choose them far apart for the best accuracy.(−3, 6) and (3, 0)
- Draw a right-angled triangle under the line and count the run. The run is the distance across, moving to the right.run = 3 − (−3) = 6
- Count the rise. The rise is the distance up (positive) or down (negative) while moving right along the line.rise = 0 − 6 = -6
- Mark the triangle on the graph.
- Divide the rise by the run. Gradient = rise ÷ run, simplified.gradient = -6 ÷ 6 = −1
Check. The line goes down from left to right, so the gradient should be negative. Also, for every 1 step across the line goes down 1. ✓
Watch out. If the line goes down as you move right, the rise is negative. Always read the line from left to right.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run. Draw a triangle on the line and count the squares up (or down) and across.
- Choose two points on the line that are exactly on grid corners. Points that sit on crossing grid lines are easy to read accurately. Choose them far apart for the best accuracy.(−6, −4) and (0, −1)
- Draw a right-angled triangle under the line and count the run. The run is the distance across, moving to the right.run = 0 − (−6) = 6
- Count the rise. The rise is the distance up (positive) or down (negative) while moving right along the line.rise = −1 − (−4) = 3
- Mark the triangle on the graph.
- Divide the rise by the run. Gradient = rise ÷ run, simplified.gradient = 3 ÷ 6 = 12
Check. The line goes up from left to right, so the gradient should be positive. Also, for every 1 step across the line goes up 12. ✓
Watch out. If the line goes down as you move right, the rise is negative. Always read the line from left to right.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run. Draw a triangle on the line and count the squares up (or down) and across.
- Choose two points on the line that are exactly on grid corners. Points that sit on crossing grid lines are easy to read accurately. Choose them far apart for the best accuracy.(−2, 8) and (3, −7)
- Draw a right-angled triangle under the line and count the run. The run is the distance across, moving to the right.run = 3 − (−2) = 5
- Count the rise. The rise is the distance up (positive) or down (negative) while moving right along the line.rise = −7 − 8 = -15
- Mark the triangle on the graph.
- Divide the rise by the run. Gradient = rise ÷ run, simplified.gradient = -15 ÷ 5 = −3
Check. The line goes down from left to right, so the gradient should be negative. Also, for every 1 step across the line goes down 3. ✓
Watch out. If the line goes down as you move right, the rise is negative. Always read the line from left to right.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run. Draw a triangle on the line and count the squares up (or down) and across.
- Choose two points on the line that are exactly on grid corners. Points that sit on crossing grid lines are easy to read accurately. Choose them far apart for the best accuracy.(−6, −6) and (0, −2)
- Draw a right-angled triangle under the line and count the run. The run is the distance across, moving to the right.run = 0 − (−6) = 6
- Count the rise. The rise is the distance up (positive) or down (negative) while moving right along the line.rise = −2 − (−6) = 4
- Mark the triangle on the graph.
- Divide the rise by the run. Gradient = rise ÷ run, simplified.gradient = 4 ÷ 6 = 23
Check. The line goes up from left to right, so the gradient should be positive. Also, for every 1 step across the line goes up 23. ✓
Watch out. If the line goes down as you move right, the rise is negative. Always read the line from left to right.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run = (change in y) ÷ (change in x) = (y2 − y1) ÷ (x2 − x1).
- Label the points. Call the first point (P) (x1, y1) and the second (Q) (x2, y2). It does not matter which is first, but you must be consistent.P = (1, 2), Q = (3, 8)
- Find the rise: the change in y. Rise is how far the line goes up (positive) or down (negative). Subtract the y-coordinates, second minus first.rise = y2 − y1 = 8 − 2 = 6
- Find the run: the change in x. Run is how far across to the right. Subtract the x-coordinates in the same order as the y-coordinates.run = x2 − x1 = 3 − 1 = 2
- Divide the rise by the run and simplify. Gradient = rise ÷ run. Cancel the fraction if you can. A negative divided by a positive (or the reverse) is negative.gradient = rise ÷ run = 6 ÷ 2 = 3
Check. Swap the points: (2 − 8) ÷ (1 − 3) = −6 ÷ (−2) = 3, the same. The gradient is positive: the line goes up from left to right. ✓
Watch out. Subtract in the same order on the top and the bottom. Mixing the order (y2 − y1 over x1 − x2) gives the wrong sign.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run = (change in y) ÷ (change in x) = (y2 − y1) ÷ (x2 − x1).
- Label the points. Call the first point (P) (x1, y1) and the second (Q) (x2, y2). It does not matter which is first, but you must be consistent.P = (0, 5), Q = (4, 1)
- Find the rise: the change in y. Rise is how far the line goes up (positive) or down (negative). Subtract the y-coordinates, second minus first.rise = y2 − y1 = 1 − 5 = −4
- Find the run: the change in x. Run is how far across to the right. Subtract the x-coordinates in the same order as the y-coordinates.run = x2 − x1 = 4 − 0 = 4
- Divide the rise by the run and simplify. Gradient = rise ÷ run. Cancel the fraction if you can. A negative divided by a positive (or the reverse) is negative.gradient = rise ÷ run = −4 ÷ 4 = −1
Check. Swap the points: (5 − 1) ÷ (0 − 4) = 4 ÷ (−4) = −1, the same. The gradient is negative: the line goes down from left to right. ✓
Watch out. Subtract in the same order on the top and the bottom. Mixing the order (y2 − y1 over x1 − x2) gives the wrong sign.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run = (change in y) ÷ (change in x) = (y2 − y1) ÷ (x2 − x1).
- Label the points. Call the first point (P) (x1, y1) and the second (Q) (x2, y2). It does not matter which is first, but you must be consistent.P = (−2, −1), Q = (2, 5)
- Find the rise: the change in y. Rise is how far the line goes up (positive) or down (negative). Subtract the y-coordinates, second minus first.rise = y2 − y1 = 5 − (−1) = 6
- Find the run: the change in x. Run is how far across to the right. Subtract the x-coordinates in the same order as the y-coordinates.run = x2 − x1 = 2 − (−2) = 4
- Divide the rise by the run and simplify. Gradient = rise ÷ run. Cancel the fraction if you can. A negative divided by a positive (or the reverse) is negative.gradient = rise ÷ run = 6 ÷ 4 = 32
Check. Swap the points: (−1 − 5) ÷ (−2 − 2) = −6 ÷ (−4) = 32, the same. The gradient is positive: the line goes up from left to right. ✓
Watch out. Subtract in the same order on the top and the bottom. Mixing the order (y2 − y1 over x1 − x2) gives the wrong sign.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run = (change in y) ÷ (change in x) = (y2 − y1) ÷ (x2 − x1).
- Label the points. Call the first point (P) (x1, y1) and the second (Q) (x2, y2). It does not matter which is first, but you must be consistent.P = (−3, 4), Q = (1, −4)
- Find the rise: the change in y. Rise is how far the line goes up (positive) or down (negative). Subtract the y-coordinates, second minus first.rise = y2 − y1 = −4 − 4 = −8
- Find the run: the change in x. Run is how far across to the right. Subtract the x-coordinates in the same order as the y-coordinates.run = x2 − x1 = 1 − (−3) = 4
- Divide the rise by the run and simplify. Gradient = rise ÷ run. Cancel the fraction if you can. A negative divided by a positive (or the reverse) is negative.gradient = rise ÷ run = −8 ÷ 4 = −2
Check. Swap the points: (4 − (−4)) ÷ (−3 − 1) = 8 ÷ (−4) = −2, the same. The gradient is negative: the line goes down from left to right. ✓
Watch out. Subtract in the same order on the top and the bottom. Mixing the order (y2 − y1 over x1 − x2) gives the wrong sign.
Key words. The gradient of a line is how steep it is: rise (the change up or down) divided by run (the change across, to the right). A line sloping up from left to right has a positive gradient; sloping down has a negative gradient.
Method. Gradient = rise ÷ run = (change in y) ÷ (change in x) = (y2 − y1) ÷ (x2 − x1).
- Label the points. Call the first point (P) (x1, y1) and the second (Q) (x2, y2). It does not matter which is first, but you must be consistent.P = (−3, 2), Q = (4, 2)
- Find the rise: the change in y. Rise is how far the line goes up (positive) or down (negative). Subtract the y-coordinates, second minus first.rise = y2 − y1 = 2 − 2 = 0
- Find the run: the change in x. Run is how far across to the right. Subtract the x-coordinates in the same order as the y-coordinates.run = x2 − x1 = 4 − (−3) = 7
- Divide the rise by the run and simplify. Gradient = rise ÷ run. Cancel the fraction if you can. A negative divided by a positive (or the reverse) is negative.gradient = rise ÷ run = 0 ÷ 7 = 0
Check. Swap the points: (2 − 2) ÷ (−3 − 4) = 0 ÷ (−7) = 0, the same. The gradient is zero: the line is horizontal. ✓
Watch out. Subtract in the same order on the top and the bottom. Mixing the order (y2 − y1 over x1 − x2) gives the wrong sign.
Method. The gradient is the rise divided by the run. Work them out for the two points.
- Find the change in x (the run). Subtract the x-coordinates.3 − 3 = 0
- Try to divide. Gradient = rise ÷ run = (change in y) ÷ 0.Dividing by 0 is impossible.
- Say what it means. The line goes straight up, so it is vertical.A vertical line has no gradient (the gradient is undefined).
Check. A vertical line cannot be measured as “up for each step across” because it never steps across. ✓
Watch out. The gradient of a vertical line is not 0 and not infinity in GCSE: say “undefined”.
Method. Gradient = rise ÷ run, written as a fraction and simplified. Percentage = decimal × 100.
- Write the gradient as a fraction. Rise ÷ run, both in metres.12 ÷ 150 = 12150
- Simplify. Divide the top and bottom by 6.12150 = 225
- Write as a decimal and percentage. 2 ÷ 25 = 0.08, and 0.08 × 100 = 8%.gradient = 225 = 0.08 = 8%
Check. 8% means 8 m of climb for every 100 m across: 150 m across gives 12 m. ✓
Watch out. Check that rise and run are in the same units.
Method. Work out the gradient of the ramp and compare it with the maximum allowed gradient. To compare fractions, write them with the same denominator.
- Gradient of the ramp. Rise 0.4 m, run 5 m.0.4 ÷ 5 = 0.08 = 225
- Compare with the maximum, 112. The lowest common multiple of 25 and 12 is 300. Write both with a denominator of 300.225 = 24300 and 112 = 25300
- Decide. The ramp is allowed if its gradient is no bigger than the maximum.24300 < 25300, so the ramp is not steeper than the maximum: it is allowed.
Check. As decimals: 0.08 and 1 ÷ 12 = 0.0833..., and 0.08 < 0.0833. ✓
Watch out. Compare gradients as numbers, not as the heights alone.
Method. Use gradient = rise ÷ run with the unknown k, then solve the equation.
- Write the rise and the run. Rise = change in y, run = change in x. The points are (2, k) and (6, 11).rise = 11 − k, run = 6 − 2 = 4
- Put the gradient into the formula. The gradient is 2.(11 − k) ÷ 4 = 2
- Multiply both sides by 4. This removes the division.11 − k = 8
- Solve for k. Subtract 11 from both sides, then change the sign. Or: k = 11 − 8.k = 11 − 8 = 3
Check. Gradient between (2, 3) and (6, 11): (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2. ✓
Watch out. Keep the subtraction in the same order on the top and on the bottom.
Method. Work out both gradients and compare them. The line with the larger gradient (ignoring the sign) is the steeper.
- Gradient of the first line. Rise ÷ run, from (0, 0) to (2, 5).5 ÷ 2 = 2.5
- Gradient of the second line. From (0, 0) to (3, 7).7 ÷ 3 = 2.33...
- Compare. A bigger gradient means a steeper line.2.5 > 2.33..., so the line through (2, 5) is steeper.
Check. Common denominator: 52 = 156 and 73 = 146, and 156 > 146. ✓
Watch out. Do not compare the rises or the runs alone: compare the gradients.
Method. Check Ben’s fraction: the gradient is rise ÷ run, which is (change in y) over (change in x).
- Work out the rise and the run. Subtract the coordinates in the same order.rise = 8 − 2 = 6, run = 3 − 1 = 2
- See what Ben did. He wrote 2/6, which is run ÷ rise: the wrong way up.Ben: 2/6 = 1/3
- Do it correctly. Gradient = rise ÷ run.6 ÷ 2 = 3
Check. From (1, 2) the line goes up 3 for each 1 across, so it reaches (2, 5) and (3, 8). ✓
Watch out. Rise (up) is the numerator; run (across) is the denominator.
Method. Points are on one straight line if the gradient between the first and second point equals the gradient between the second and third.
- Gradient from A to B. Rise ÷ run.(5 − 1) ÷ (2 − 0) = 4 ÷ 2 = 2
- Gradient from B to C. Use the same method for the next pair.(11 − 5) ÷ (5 − 2) = 6 ÷ 3 = 2
- Compare. Both gradients are 2, and B is a shared point, so the whole path is one straight line.AB and BC have the same gradient and share the point B, so A, B and C lie on a straight line.
Check. The gradient from A to C is (11 − 1) ÷ (5 − 0) = 10 ÷ 5 = 2 as well. ✓
Watch out. You must use shared point B: equal gradients and a common point means one line (not two parallel lines).
Method. Use gradient = rise ÷ run with the unknown k, then solve the equation by cross-multiplying.
- Write rise and run. The points are (1, 3) and (k, 9).rise = 9 − 3 = 6, run = k − 1
- Put in the gradient. The gradient is 32.6 ÷ (k − 1) = 32
- Cross-multiply. Multiply both sides by 2 and by (k − 1) to clear the fractions.6 × 2 = 3(k − 1), so 12 = 3(k − 1)
- Divide both sides by 3. 4 = k − 1
- Add 1 to both sides. k = 5
Check. Gradient between (1, 3) and (5, 9): 6 ÷ 4 = 32. ✓
Watch out. Be careful to keep k − 1 as one bracket when you multiply.
Method. The steepness of a line depends on the size of the gradient, not its sign: a negative gradient of −5 is steeper than a positive gradient of 3.
- List the sizes (ignoring the sign). The sign shows whether the line goes up or down, but the size shows the steepness.3, 5, 12, 1
- Find the largest size. 5 is the biggest number.size 5 belongs to the gradient −5
- Answer. The line with gradient −5 is the steepest (it goes steeply down).
Check. Gradient −5 means down 5 for each 1 across, which is steeper than up 3 for each 1 across. ✓
Watch out. Compare sizes, not signs: −5 is the “smallest” number but the steepest line.