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Gradient and intercept

y = mx + c and parallel lines. Topic 10 of 27 in Algebra.

Foundation & Higher This topic appears on both Foundation and Higher papers.

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Every straight line can be described with two numbers:

  • its gradient: how steep it is
  • its y-intercept: where it crosses the y-axis

This page shows you how to find both, and how to use them to write the equation of a line.

Before you start, make sure you can plot and read coordinates, work with negative numbers, simplify fractions and rearrange a simple equation.

Blue is for the gradient Red is for the y-intercept

1. What gradient means

Key idea Gradient is how steep a line is. Gradient = rise ÷ run

The gradient tells you how far a line goes up or down for every 1 you move to the right.

  • Line goes uphill (left to right): gradient is positive.
  • Line goes downhill (left to right): gradient is negative.
  • Line is flat: gradient is 0.
  • The bigger the number (ignoring the sign), the steeper the line. A gradient of −5 is steeper than a gradient of 2.

To measure it, pick two points on the line.

  • The rise is how far you go up or down between them.
  • The run is how far you go across.

gradient = riserun = change in ychange in x

A straight line has the same gradient everywhere. Any two points on it give the same answer.

−6−5−4−3−2−1123456−6−5−4−3−2−11234560xyrun = +6rise = +4A(−4, −1)B(2, 3)

2. Finding the gradient from a graph

Method

  1. Pick two points on the line
    Choose points where the line goes exactly through a corner of the grid. Far apart is easier and more accurate.
  2. Draw a right-angled triangle between them
    Read the rise (up is +, down is −) and the run (to the right is +).
  3. Divide the rise by the run, then simplify
Watch out Check the scale on each axis. Count in the axis values, not in squares.

Example 1: a positive gradient

Find the gradient of this line.

−6−5−4−3−2−1123456−6−5−4−3−2−11234560xy6 across4 up(−4, −1)(2, 3)
  1. Pick two points on grid corners
    (−4, −1) and (2, 3)
  2. Read the rise and run
    rise = 4 up    run = 6 across
  3. Divide the rise by the run, then simplify
    gradient = 46 = 23

The line goes uphill, so the gradient should be positive. It is.

Answergradient = 23

Example 2: a negative gradient

Find the gradient of this line.

−6−5−4−3−2−1123456−6−5−4−3−2−11234560xy8 across6 down(−4, 3)(4, −3)
  1. Pick two points on grid corners
    (−4, 3) and (4, −3)
  2. Read the rise and run, going left to right
    rise = 6 down = −6    run = 8 across
    Down is negative.
  3. Divide the rise by the run, then simplify
    gradient = −68 = −34

The line goes downhill, so the gradient should be negative. It is.

Answergradient = −34

Example 3: axes with different scales

Find the gradient of this line.

Look at the axes first: the x-axis goes up in 1s, the y-axis goes up in 10s.

12345020406080100xy4 across60 up(0, 20)(4, 80)
  1. Read two points from the axes
    (0, 20) and (4, 80)
  2. Work out the rise and run from the values
    rise = 80 − 20 = 60
    run = 4 − 0 = 4
  3. Divide the rise by the run
    gradient = 604 = 15
Answergradient = 15
Watch out Counting squares gives 6 up and 4 across, so 1.5. That’s wrong, because each square up the y-axis is worth 10.

3. Finding the gradient from two coordinates

Key idea No graph? Use the formula. It’s still change in y ÷ change in x.

Call the points (x1, y1) and (x2, y2). The letter m stands for the gradient.

m = y2 − y1x2 − x1

  • Either point can be first, but use the same order on the top and the bottom.
  • Put negative numbers in brackets, so the signs don’t get lost.

Example 4: two coordinates

Find the gradient of the line through A(1, −3) and B(5, 7).

  1. Label the points
    x1 = 1, y1 = −3    x2 = 5, y2 = 7
  2. Put them into the formula
    m = 7 − (−3)5 − 1
  3. Work out the top and the bottom
    top: 7 − (−3) = 7 + 3 = 10
    bottom: 5 − 1 = 4
    Subtracting a negative is the same as adding.
  4. Simplify
    m = 104 = 52
Answerm = 52

Example 5: negative coordinates and a negative gradient

Find the gradient of the line through C(−2, 9) and D(4, −6).

  1. Put the points into the formula
    m = −6 − 94 − (−2)
  2. Work out the top and the bottom
    top: −6 − 9 = −15
    bottom: 4 − (−2) = 4 + 2 = 6
  3. Simplify
    m = −156 = −52

Quick sketch: D is to the right of C and lower down, so the line goes downhill. A negative gradient makes sense.

Answerm = −52

4. Horizontal and vertical lines

Key idea y = a number is flat. x = a number goes straight up and down.
LineDirectionEvery point on it hasGradient
y = −3horizontal (flat)y-coordinate −3, e.g. (−5, −3), (0, −3), (6, −3)0: the rise is 0
x = 4vertical (straight up)x-coordinate 4, e.g. (4, −5), (4, 0), (4, 6)none you can work out: the run is 0, and you can’t divide by 0

The x-axis is the line y = 0. The y-axis is the line x = 0.

Watch out It’s easy to mix these up. x = 4 is vertical, even though it has an x in it: every point on it has an x-coordinate of 4.
−6−5−4−3−2−1123456−6−5−4−3−2−11234560xyy = −3x = 4

5. The y-intercept and y = mx + c

Key idea In y = mx + c, m is the gradient and c is the y-intercept.

The y-intercept is where a line crosses the y-axis. On the y-axis, x is always 0, so the y-intercept is a point like (0, 5) or (0, −2).

Every straight line, except a vertical one, has an equation like this:

y = mx + c

  • m is the gradient.
  • c is the y-intercept: the line crosses the y-axis at (0, c).

Why it works:

  • Put x = 0 into y = mx + c and you get y = c. So the line crosses the y-axis at c.
  • Each time x goes up by 1, the mx part goes up by m. That’s the gradient.
−6−5−4−3−2−1123456−6−5−4−3−2−11234560xy3 across2 up(0, −1)

6. Reading m and c from an equation

Key idea Get the equation to start y = first. Then read off m and c.

The sign in front of a number belongs to that number.

Example 6: already in the form y = mx + c

Find the gradient and y-intercept of y = 4x − 7.

y = 4x − 7

Answerm = 4,  c = −7

So the line crosses the y-axis at (0, −7).

Example 7: the x term comes second

Find the gradient and y-intercept of y = 6 − x.

  1. Swap the order so the x term comes first
    y = −x + 6
  2. −x on its own means −1x
    y = −1x + 6
Answerm = −1,  c = 6

Example 8: rearrange first

Find the gradient and y-intercept of 5x + 2y = 8.

  1. Get the y term on its own: subtract 5x from both sides
    2y = −5x + 8
  2. Divide every term by 2
    y = −52x + 4
Answerm = −52,  c = 4
Watch outThe gradient is not 5.

If there are brackets, expand them first. y = 3(x − 2) is y = 3x − 6, so m = 3 and c = −6.

7. Finding the equation of a line

Key idea To write the equation of a line, find m, then find c, then write y = mx + c.

A question can give you the line in three ways: as a graph, as a gradient and one point, or as two points.

Example 9: from a graph

Find the equation of this line.

−6−5−4−3−2−1123456−6−5−4−3−2−11234560xy(0, 3)
  1. Find c: where does the line cross the y-axis?
    c = 3
  2. Find m: it goes down 1 for every 2 across
    m = −12
  3. Put them into y = mx + c
Answery = −12x + 3

Example 10: from the gradient and one point

A line has gradient −3 and passes through (4, −5). Find its equation.

  1. Write what you know
    y = −3x + c
  2. Put in the point: x = 4, y = −5
    −5 = −3 × 4 + c
  3. Tidy up
    −5 = −12 + c
  4. Solve for c: add 12 to both sides
    c = 7
Answery = −3x + 7

Example 11: from two points

Find the equation of the line through (−2, −5) and (4, 4).

  1. Find the gradient first
    m = 4 − (−5)4 − (−2) = 96 = 32
  2. Put either point in. Using (−2, −5):
    −5 = 32 × (−2) + c
  3. Tidy up
    −5 = −3 + c
  4. Solve for c: add 3 to both sides
    c = −2
Answery = 32x − 2

Check with the other point, (4, 4): 32 × 4 − 2 = 6 − 2 = 4. Correct.

Watch out When a question asks for the equation, the answer is the whole equation, y = …, not just the values of m and c.
−6−5−4−3−2−1123456−6−5−4−3−2−11234560xyrun = +6rise = +9P(−2, −5)Q(4, 4)

8. Parallel lines

Key idea Parallel lines have the same gradient.

They point the same way, so they never meet. Their values of c are different. If c were the same too, they would be the same line.

To check whether two lines are parallel, rearrange both into y = mx + c and compare the values of m.

Example 12: which lines are parallel?

Which of these lines are parallel to y = 3x + 1?

LineAs y = mx + cGradientParallel?
y = 3x − 5already is3Yes
y = 1 − 3xy = −3x + 1−3No
2y = 6x + 7divide by 2: y = 3x + 3.53Yes
y = 13x + 1already is13No, even though c is the same

Example 13: a parallel line through a point

Find the equation of the line parallel to y = 5x − 2 that passes through (1, 9).

  1. Parallel, so the gradient is the same
    y = 5x + c
  2. Put in the point: x = 1, y = 9
    9 = 5 × 1 + c
  3. Solve for c
    9 = 5 + c, so c = 4
Answery = 5x + 4
−6−5−4−3−2−1123456−6−5−4−3−2−11234560xyAB

9. What m and c mean in real life

Key idea On a real-life graph, the intercept is the starting value and the gradient is the rate.
  • The intercept is the value when the x quantity is 0: often a starting value or a fixed charge.
  • The gradient is how much the y quantity goes up for each 1 of the x quantity: a rate. Its units are “y units per x unit”, such as pounds per hour.

Example 14: a plumber’s charges

The graph shows how much a plumber charges, C pounds, for a job lasting h hours. What do the intercept and gradient tell you?

1234504080120160200240(0, 40)(4, 180)Time, h (hours)Cost, C (£)
  1. Read the intercept
    The line meets the cost axis at 40.
    The plumber charges £40 before any work is done: a call-out fee.
  2. Work out the gradient
    gradient = 180 − 404 − 0 = 1404 = 35
    The plumber charges £35 for each hour.
  3. Write the equation
AnswerC = 35h + 40

The mistakes that cost marks

  1. Dividing the wrong way round. Across over up gives the wrong answer. rise ÷ run: up over across.
  2. Mixing the order of the points. It flips the sign. if y2 comes first on the top, x2 comes first on the bottom.
  3. Counting squares when the scales are different. read the values on each axis, then subtract.
  4. Losing the minus sign. look at the line before you write the answer. Downhill means negative.
  5. Reading m from an equation that doesn’t start y =. The gradient of 5x + 2y = 8 is not 5. rearrange to y = … first.
  6. Dropping the sign of c. y = 4x − 7 crosses the y-axis at −7, not 7. keep the sign in front of the number with it.
  7. Mixing up x = k and y = k. remember x = 4 is a vertical line and y = 4 is a horizontal line.
  8. Thinking lines with the same c are parallel. The same c means they cross the y-axis at the same point. check the gradients. Parallel lines have the same m.
  9. Stopping at m and c. if the question asks for the equation, write y = … in full.

Quick checklist

  • I can say whether a line’s gradient is positive, negative or 0 just by looking at it.
  • I can find a gradient from a graph, and I check the scale on both axes first.
  • I can find the gradient between two coordinates, including negative ones.
  • I know which way the lines x = k and y = k go.
  • I can read m and c from y = mx + c, including equations like y = 6 − x.
  • I can rearrange an equation like 5x + 2y = 8 into y = mx + c.
  • I can find the equation of a line from a graph, from a gradient and a point, and from two points.
  • I know parallel lines have the same gradient, and I can find a parallel line through a given point.
  • I can explain what the gradient and intercept mean on a real-life graph.

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