Gradient is the single number that says how steep a line is and which way it leans. Every form of a linear equation is just a different way of displaying that number and one point.
Today's hook, A straight line is the only graph completely fixed by two numbers. Get the gradient and one point and you have the whole line, forever, in both directions.
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Recall, your gut answer first
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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Recall, your gut answer first
+5 XP warm-up
A ramp rises 1 m over a horizontal run of 12 m. Another rises 2 m over 30 m. Which is steeper, and what single number would let you compare them without drawing anything?
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
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Gradient is rise over run
Work through the core explanation before applying it.
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Gradient is rise over run
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Gradient measures how much $y$ changes for each unit increase in $x$: $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$. A positive gradient leans up to the right, a negative one leans down, and zero is horizontal.
$y = mx + b$ gradient $m$, $y$-intercept $b$ $ax + by + c = 0$ general form
Gradient-intercept form reads straight off
In $y = 3x - 5$ the gradient is 3 and the $y$-intercept is $-5$. No work required, provided $y$ is alone on the left.
Rearrange before you read
$2y = 6x + 8$ is not in gradient-intercept form. Divide by 2 first to get $y = 3x + 4$, or you will read the gradient as 6.
General form has zero on the right
$ax + by + c = 0$, with $a$, $b$ and $c$ usually integers and $a$ positive. It is the form to use when a question says "in general form".
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
Gradient is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$, the change in $y$ per unit change in $x$.
Gradient-intercept form is $y = mx + b$, where $m$ is the gradient and $b$ the $y$-intercept.
General form is $ax + by + c = 0$, with the right-hand side zero.
A horizontal line has gradient 0; a vertical line has no gradient, because the run is zero.
Understand
Concepts
Why gradient is a rate of change, not just a measure of steepness.
Why $y$ must be isolated before the gradient can be read off.
Why a vertical line has no gradient rather than a gradient of zero.
Can do
Skills
Calculate a gradient from two points or from a graph.
Read gradient and intercept from gradient-intercept form, rearranging first if needed.
Convert between gradient-intercept form and general form in both directions.
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Key terms
GradientA number saying how much the line rises for each unit it runs right. Like this: a gradient of 3 means the line goes up 3 units every time $x$ increases by 1.
Rise and runThe vertical and horizontal changes between two points on a line. Like this: from $(1,2)$ to $(4,11)$ the rise is 9 and the run is 3, so the gradient is 3.
$y$-interceptThe $y$-value where the line crosses the $y$-axis, which happens when $x = 0$. Like this: $y = 3x - 5$ crosses at $(0, -5)$, so the $y$-intercept is $-5$.
Gradient-intercept formA linear equation written as $y = mx + b$, so the gradient and intercept can be read directly. Like this: $y = -2x + 7$ has gradient $-2$ and $y$-intercept 7.
General formA linear equation written as $ax + by + c = 0$, with everything on one side. Like this: $y = 3x + 4$ becomes $3x - y + 4 = 0$.
Undefined gradientWhat a vertical line has, because its run is zero and you cannot divide by zero. Like this: $x = 4$ is vertical, so its gradient is undefined, not zero.
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Calculating a gradient
Work through the core explanation before applying it.
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Calculating a gradient
core concept
Take any two points on the line, $(x_1, y_1)$ and $(x_2, y_2)$. The gradient is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$, the rise divided by the run.
From $(1, 2)$ to $(4, 11)$: $m = \dfrac{11 - 2}{4 - 1} = \dfrac{9}{3} = 3$. Every step right of 1 lifts the line 3.
It does not matter which point you call first, as long as you subtract in the same order top and bottom. Reversing both gives $\dfrac{2 - 11}{1 - 4} = \dfrac{-9}{-3} = 3$, the same answer.
Subtract in the same order. $\dfrac{y_2 - y_1}{x_1 - x_2}$ gives the negative of the right answer. Mismatching the order is the most common gradient error.
Quick check: what is the gradient of the line through $(2, 5)$ and $(6, 13)$?
Gradient $m = \frac{y_2-y_1}{x_2-x_1}$, rise over run. Subtract the coordinates in the same order top and bottom. A positive gradient rises to the right, a negative one falls, zero is horizontal, and a vertical line has an undefined gradient.
Pause, copy the gradient formula, the worked $(1,2)$ to $(4,11)$ giving 3, and the same-order warning, into your book.
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Gradient-intercept form
core concept
We just saw how to get a gradient from two points. That raises a question: can you read the gradient straight out of an equation, without finding any points at all? This card answers it → yes, provided the equation is arranged with $y$ by itself.
When a line is written $y = mx + b$, the coefficient of $x$ is the gradient and the constant is the $y$-intercept. So $y = -2x + 7$ has gradient $-2$ and crosses the $y$-axis at 7.
The form only works when $y$ is alone. $2y = 6x + 8$ is not ready: divide by 2 to get $y = 3x + 4$, so the gradient is 3, not 6.
To build an equation, substitute the two numbers. Gradient $\tfrac{1}{2}$ and $y$-intercept $-3$ gives $y = \tfrac{1}{2}x - 3$.
Check the sign of the constant. $y = 4x - 9$ has $y$-intercept $-9$, not 9. The form is $mx + b$, so a subtraction means a negative $b$.
Fill the blank: the gradient of $3y = 12x - 6$ is .
In $y = mx + b$ the coefficient of $x$ is the gradient and the constant is the $y$-intercept, but only once $y$ is alone on the left. Rearrange first. A minus sign in front of the constant means a negative intercept.
Pause, copy the form $y = mx + b$, the rearrangement of $2y = 6x + 8$ to $y = 3x + 4$, and the sign warning, into your book.
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General form, and converting between the two
core concept
We just saw that gradient-intercept form displays the two numbers you need. That raises a question: why does the syllabus also want $ax + by + c = 0$? This card answers it → general form handles every line, including vertical ones, and is the standard way to present an answer with integer coefficients.
General form is $ax + by + c = 0$: everything on the left, zero on the right, coefficients usually integers with $a$ positive.
To convert from gradient-intercept form, move every term to one side. $y = 3x + 4$ becomes $0 = 3x - y + 4$, written $3x - y + 4 = 0$.
To convert back, solve for $y$. From $2x + 5y - 10 = 0$: $5y = -2x + 10$, so $y = -\tfrac{2}{5}x + 2$, giving gradient $-\tfrac{2}{5}$ and $y$-intercept 2.
General form covers vertical lines. $x = 4$ has no gradient-intercept form at all, but in general form it is $x - 4 = 0$. That is why the syllabus asks for both.
Which equation is NOT in general form?
General form is $ax + by + c = 0$, everything on one side with integer coefficients and $a$ positive. Convert to it by moving all terms left; convert back by solving for $y$. General form can express a vertical line, which gradient-intercept form cannot.
Pause, copy general form, both conversions worked in each direction, and the note that $x = 4$ has only a general form, into your book.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · GRADIENT FROM TWO POINTS
Find the gradient of the line through $(-2, 7)$ and $(4, -5)$.
Careful with the double negative in the denominator: $4 - (-2) = 6$.
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$m = -2$
Negative, so the line falls as $x$ increases.
PROBLEM 2 · READING A REARRANGED EQUATION
Find the gradient and $y$-intercept of $4y - 12x = 20$.
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$4y = 12x + 20$
Move the $x$ term to the right.
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$y = 3x + 5$
Divide every term by 4 so $y$ is alone.
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Gradient 3, $y$-intercept 5
Now the form is $y = mx + b$ and both read straight off.
PROBLEM 3 · CONVERTING TO GENERAL FORM
Write $y = -\tfrac{3}{4}x + 2$ in general form with integer coefficients.
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Multiply through by 4: $4y = -3x + 8$
Clear the fraction first, or the coefficients will not be integers.
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$3x + 4y - 8 = 0$
Move everything to the left so the right-hand side is zero.
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Check $a > 0$: yes, $a = 3$
Convention is a positive leading coefficient.
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Quick-fire practice
Work through the core explanation before applying it.
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Quick-fire practice
+10 XP
Find the gradient through $(0, 1)$ and $(3, 10)$.
State the gradient and $y$-intercept of $y = -5x + 2$.
Find the gradient of $2y = 8x - 6$.
Write $y = 2x - 7$ in general form.
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Revisit the two ramps
Run the quick drill and copy the summary into your book.
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Revisit the two ramps
At the start you compared a 1-in-12 ramp with a 2-in-30 ramp. Give both gradients as decimals, say which is steeper, and explain why gradient is the right single number for the comparison.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
+5 XP per correct · +25 XP all-correct
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Short answer
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Short answer
ApplyBand 43 marks
Q1. Find the gradient and $y$-intercept of the line $6y - 9x = 24$, and hence state its equation in gradient-intercept form. (3 marks)
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ApplyBand 43 marks
Q2. Write the equation of the line with gradient $-\tfrac{2}{3}$ and $y$-intercept 5 in general form with integer coefficients. (3 marks)
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UnderstandBand 32 marks
Q3. Explain why a vertical line has an undefined gradient rather than a gradient of zero, and state what a gradient of zero does describe. (2 marks)
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📖 Comprehensive answers (click to reveal)
Practice 1: $m = \frac{10-1}{3-0} = 3$. Practice 2: gradient $-5$, $y$-intercept 2. Practice 3: $y = 4x - 3$, gradient 4. Practice 4: $2x - y - 7 = 0$.
Q3 (2 marks): A vertical line has run zero, and gradient is rise divided by run, so the division is by zero and undefined [1]. A gradient of zero describes a horizontal line, where the rise is zero for any run [1].
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Review and finish
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Boss battle · Gradient Grind
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Read, calculate and convert gradients and intercepts at speed. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.