The equation of a straight line
Introduced| English | Meaning |
|---|---|
| intercept/ˌɪntəˈsept/ | The coordinate where a graph meets a specified axis, including where it just touches. |
| line of best fit/laɪn ɒv best fɪt/ | A line chosen to summarise the overall relationship between two variables in a scatter plot. |
Reading a line like a barcode
- Every straight line has a unique "signature" — its equation.
- If you can read the equation, you can tell the gradient, the intercept 截距, and where the line crosses every axis — without drawing it.
- For a non-vertical line, use $y=mx+c$. A vertical line instead has equation $x=k$.
The form $y = mx + c$
- $m$ is the gradient (how steep the line is).
- $c$ is the $y$-intercept (where the line crosses the $y$-axis — the point $(0, c)$).
- Together, they describe a non-vertical line. A horizontal line has $m=0$; a vertical line has no finite gradient.
In $y = 3x - 1$: the gradient $m = 3$ and the intercept $c = -1$. The figure shows both.
The straight line
y = ax + b
Every line has a gradient a and a y-intercept b.
In y = mx + c, the letter m stands for the:
Difficulty 1/5 · 100 coins + a small bonus on first completion today
m is the gradient (steepness); c is the y-intercept.
Reading $m$ and $c$ from an equation
- When the equation is already in $y = mx + c$ form, just read off the values:
- $y = 5x + 2$ → $m = 5$, $c = 2$.
- $y = -\dfrac{1}{2}x + 4$ → $m = -\dfrac{1}{2}$, $c = 4$.
- $y = 3x$ → $m = 3$, $c = 0$ (the line passes through the origin).
Extended: not in $y =$ form? You must rearrange first. $5x + 4y = 8$ looks tricky, but: $4y = -5x + 8$, so $y = -\dfrac{5}{4}x + 2$ — gradient $-\dfrac{5}{4}$, intercept $2$.
For $y=mx+c$ the line crosses the $y$-axis at $c$, and the gradient $m$ is the rise divided by the run
Finding the equation from a point and gradient
- Know the gradient $m$ and one point? Substitute into $y = mx + c$ and solve for $c$.
- Example: gradient $3$ through $(1, 2)$:
- $2 = 3(1) + c \Rightarrow c = 2 - 3 = -1$.
- The equation is $y = 3x - 1$.
Extended: from two given coordinate pairs. Through $(2, 5)$ and $(4, 11)$: first find $m = \dfrac{11-5}{4-2} = \dfrac{6}{2} = 3$. Then $5 = 3(2) + c \Rightarrow c = -1$. Equation: $y = 3x - 1$.
A line has gradient 3 and passes through (1, 2). In y = 3x + c, what is c?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
2 = 3(1) + c, so c = 2 − 3 = −1; the line is y = 3x − 1.
A line has gradient 2 and passes through (3, 1). Its equation is:
Difficulty 1/5 · 100 coins + a small bonus on first completion today
1 = 2(3) + c → c = 1 − 6 = −5. So y = 2x − 5.
A line has gradient −1/2 and passes through (4,1). In y=mx+c, what is c?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
1=(−1/2)(4)+c gives 1=−2+c, so c=3. This uses a given gradient; perpendicular-line methods come next.
Extended: find the equation through (−2,5) and (4,−4). Give it in fully simplified y=mx+c form and check both points.
A coordinate-pair calculation without a supplied grid is Extended.
Rearranging to $y = mx + c$ (Extended)
- Exams love giving equations like $ax + by = c$ and asking for the gradient.
- Step 1: isolate $y$ — move $ax$ to the other side, then divide by $b$.
- Step 2: read off $m$ and the intercept.
- Example: $2x + 3y = 12 \Rightarrow 3y = -2x + 12 \Rightarrow y = -\dfrac{2}{3}x + 4$.
- Gradient $= -\dfrac{2}{3}$; $y$-intercept $= 4$.
Extended: Rearrange 5x + 4y = 8 into y = mx + c. What is c (the y-intercept)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
4y = −5x + 8, so y = −(5/4)x + 2; the intercept is 2.
Extended: Rearrange 2x + 3y = 12 into y = mx + c. What is the gradient m?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
3y = −2x + 12, so y = −(2/3)x + 4. The gradient is −2/3 ≈ −0.667.
Extended: Rearrange 3x − y = 7 into y = mx + c. What is the y-intercept?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
−y = −3x + 7, so y = 3x − 7. The y-intercept is −7.
Extended: rearrange 6x−4y=12 into y=mx+c. State the gradient and both intercepts. Find the equation of the line through (2,−1) with that same gradient.
The old intercept does not transfer to a different parallel line.
The line of best fit 最佳拟合线
- In science experiments, data points rarely fall on a perfect line.
- Scientists draw a line of best fit through the scatter — and its equation $y = mx + c$ summarises the whole experiment in one sentence.
- The gradient tells you the rate (e.g. temperature change per minute). The intercept is the model value when $x=0$; call it a starting value only if $x=0$ is the start and lies in the situation being modelled.
Core and Extended: equation from a graph. Read where the drawn line crosses the $y$-axis to find $c$. Use a large rise/run triangle on the grid to find $m$, then write $y=mx+c$ in simplified form. Check a different plotted point. For example, a graph crossing at $(0,-2)$ and rising $6$ units for $3$ units right gives $y=2x-2$. A vertical graph through $x=4$ is $x=4$, not $y=4$.
Use this graph for the independent graph-to-equation task.
Match each equation to its property.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
y = 4x + 1: m = 4. y = −x + 3: c = 3. y = 2x: c = 0, so it passes through (0,0). y = 5x − 1 has gradient 5, same as y = 5x + 7, so they are parallel.
Core and Extended: use graph L above. Read its y-intercept and use a gradient triangle to obtain its equation. Check another grid point. Also give the equation of the vertical line through x=−2.
An intercept is an axis-crossing coordinate; c is its y-value, not its x-value.
You've got it
- $y = mx + c$ — $m$ is the gradient, $c$ is the $y$-intercept
- Extended: rearrange $ax + by = c$ into $y = mx + c$ before reading off $m$ and $c$
- gradient $3$ through $(1, 2)$ → $c = -1$ → equation $y = 3x - 1$
- Extended, from two given coordinate pairs: find $m$ first, then substitute one point to find $c$
Rearrange independently (Extended)
- Rearrange 4x plus 2y equals 10 into y equals mx plus c.
- Give c.
Extended: Rearrange 4x plus 2y equals 10 into y equals mx plus c. What is c?
10 divided by 2 is 5.
Extended: rearrange 4x+2y=10 into y=mx+c. What is c?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
The constant term is 5. Other options — 2 (the lesson intercept); Negative 2 (the gradient).