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Lesson 11 ~30 min Unit 2 · Linear Relationships +95 XP

Equation of a Line: $y = mx + c$

Master the gradient-intercept form. Write, read and rearrange the equation of any straight line using just two numbers: $m$ and $c$.

Think about it: You know a line has gradient 2 and crosses the y-axis at $(0, 3)$. How could you write its equation in a compact form?
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Think First
warm-up

You know a line has gradient 2 and crosses the y-axis at $(0, 3)$. How could you write its equation in a compact form?

Record your answer in your workbook.
1
Introducing $y = mx + c$
+5 XP

The equation $y = mx + c$ is the most important formula for straight lines. It tells you everything you need to know to draw or describe any line.

$$y = \underbrace{m}_{\text{gradient}}x + \underbrace{c}_{\text{y-intercept}}$$
y = m x + c Gradient (slope) y-intercept Just two numbers — $m$ and $c$ — describe any straight line.
What $m$ tells us
Positive $m$: slopes up. Negative $m$: slopes down. Larger $|m|$: steeper. $m = 0$: horizontal.
What $c$ tells us
Positive $c$: crosses above origin. Negative $c$: below origin. $c = 0$: passes through origin.
EquationGradient ($m$)y-intercept ($c$)
$y = 2x + 1$$2$$1$
$y = -3x + 5$$-3$$5$
$y = x - 4$$1$$-4$
$y = \frac{1}{2}x$$\frac{1}{2}$$0$
$y = -x + 3$$-1$$3$
2
Write the Equation from $m$ and $c$
+5 XP

When you are given the gradient and y-intercept, substitute them directly into $y = mx + c$.

Worked Example 1 — Positive Values
A line has gradient $m = 2$ and y-intercept $c = 3$. Write its equation.
  1. Start with $y = mx + c$.
  2. Substitute $m = 2$ and $c = 3$: $y = 2x + 3$.
  3. The equation of the line is $y = 2x + 3$. (Gradient 2, crosses y-axis at $(0, 3)$.)
Worked Example 2 — Negative Gradient
A line has gradient $m = -1$ and y-intercept $c = 4$. Write its equation.
  1. Start with $y = mx + c$.
  2. Substitute $m = -1$ and $c = 4$: $y = (-1)x + 4$.
  3. Simplify $(-1)x$ to $-x$: the equation is $y = -x + 4$. (Slopes downward, crosses at $(0, 4)$.)
3
Finding $m$ and $c$ from an Equation
+5 XP

When an equation is already in $y = mx + c$ form, read $m$ and $c$ directly by matching terms.

Worked Example 3 — Read $m$ and $c$ Directly
Find the gradient and y-intercept of $y = 3x - 2$.
  1. Compare $y = 3x - 2$ with $y = mx + c$.
  2. The coefficient of $x$ is $3$, so $m = 3$. The constant term is $-2$, so $c = -2$.
  3. Gradient $= 3$; y-intercept $= -2$ (the line crosses the y-axis at $(0, -2)$).
x y O 1 2 3 1 2 3 $c = -2$ run = 1 rise = 3 $m = 3$ $y = 3x - 2$
Watch the sign!
In $y = 3x - 2$, the constant $c = -2$ (not $+2$). The minus sign belongs to the 3.
4
Rearranging to $y = mx + c$ Form
+5 XP

Some equations are not given in gradient-intercept form. Rearrange by making $y$ the subject.

Worked Example 4 — Subtract to Isolate $y$
Rearrange $2x + y = 5$ into $y = mx + c$ form, then state $m$ and $c$.
  1. Start: $2x + y = 5$.
  2. Subtract $2x$ from both sides: $y = 5 - 2x$.
  3. Rewrite with $x$ term first: $y = -2x + 5$.
  4. $m = -2$, $c = 5$. The gradient is $-2$ and the line crosses the y-axis at $(0, 5)$.
Divide through example
For $3y = 6x + 9$, divide every term by 3: $y = 2x + 3$. So $m = 2$, $c = 3$.
Key strategy
Get $y$ alone on the left. Move everything else to the right. Divide if the coefficient of $y$ is not 1.
5
Using the Equation to Predict and Check
+5 XP

Once you have $y = mx + c$, substitute any $x$-value to find the matching $y$-value, or check if a point lies on the line.

Predict a point
$y = 2x + 1$: when $x = 3$, $y = 2(3) + 1 = 7$. The point $(3, 7)$ lies on the line.
Check a point
Does $(2, 5)$ lie on $y = 2x + 1$? Sub in: $2(2)+1 = 5 = $ LHS. Yes! Both sides equal.
Point NOT on line
Does $(1, 4)$ lie on $y = 2x + 1$? Sub in: $2(1)+1 = 3 \neq 4$. No — the sides don't match.
6
The Big Picture — $y = mx + c$ on a Graph
+5 XP
x y O 1 2 3 4 5 1 2 3 4 5 $c = 2$ y-intercept $(0, 2)$ run = 2 rise = 1 $m = \frac{1}{2}$ $y = \frac{1}{2}x + 2$

The line $y = \frac{1}{2}x + 2$ has gradient $m = \frac{1}{2}$ and y-intercept $c = 2$.

Summary
$m$ = steepness and direction. $c$ = where the line starts on the y-axis. Together they fully define the line.
Brain Trainer — $y = mx + c$
+10 XP

Answer as quickly as you can — no calculators!

$m = 3$, $c = 5$. Write the equation.
?$y = 3x + 5$
$m = -2$, $c = 4$. Write the equation.
?$y = -2x + 4$
Gradient of $y = 5x - 3$?
?$m = 5$
y-intercept of $y = -x + 6$?
?$c = 6$
Rearrange $y - 4x = 2$.
?$y = 4x + 2$
Rearrange $x + y = 7$.
?$y = -x + 7$
$m$ and $c$ in $y = \frac{1}{3}x - 4$?
?$m = \frac{1}{3}$, $c = -4$
Through $(0,0)$, gradient $-4$.
?$y = -4x$
Rearrange $3y = 9x - 6$.
?$y = 3x - 2$
Does $(2, 7)$ lie on $y = 3x + 1$?
?$3(2)+1 = 7$. Yes!

Multiple Choice

Q1. What is the equation of a line with gradient $3$ and y-intercept $-2$?

Q2. In the equation $y = -2x + 5$, what is the gradient?

Q3. Rearrange $x + y = 3$ into the form $y = mx + c$.

Q4. A line has equation $y = \frac{1}{2}x + 3$. What are $m$ and $c$?

Q5. Which equation represents a line with gradient $-3$ passing through $(0, 2)$?

Short Answer

SAQ 1. Write the equation of a line with gradient $4$ and y-intercept $-1$. Show your substitution into $y = mx + c$.

Show answer

$y = mx + c$ with $m = 4$, $c = -1$: $y = 4x + (-1) = 4x - 1$.

SAQ 2. Rearrange $2y = 4x + 6$ into $y = mx + c$ form, then state the values of $m$ and $c$.

Show answer

$2y = 4x + 6$. Divide every term by 2: $y = 2x + 3$. So $m = 2$ and $c = 3$.

SAQ 3. Explain what $m$ and $c$ each tell you about a line. Give one example for each.

Show answer

$m$ (gradient): tells how steep the line is and which direction it slopes. Positive $m$ slopes up, negative $m$ slopes down. Example: in $y = 2x + 3$, $m = 2$ means the line rises 2 for every 1 to the right.

$c$ (y-intercept): tells where the line crosses the y-axis — the point $(0, c)$. Example: in $y = 2x + 3$, $c = 3$ means the line crosses at $(0, 3)$.

Stretch Challenges

Challenge 1 — Equation from intercept and gradient

A line passes through $(0, -3)$ and has gradient $\frac{2}{3}$. Write its equation and find the coordinates of the point where $x = 6$.

Solution

$y = mx + c$ with $m = \frac{2}{3}$, $c = -3$: $y = \frac{2}{3}x - 3$.

When $x = 6$: $y = \frac{2}{3}(6) - 3 = 4 - 3 = 1$. Point: $(6, 1)$.

Challenge 2 — Verify a point

Show that $(-2, 7)$ lies on the line $y = -3x + 1$. Then find a point that does not lie on this line.

Solution

Sub $x = -2$: $y = -3(-2) + 1 = 6 + 1 = 7$. LHS $= 7$ = RHS. ✓ The point lies on the line.

Example of a point not on the line: $(0, 2)$. Sub: $-3(0)+1 = 1 \neq 2$. ✗

Challenge 3 — Both intercepts

A line has equation $3x + 2y = 12$. Rearrange into $y = mx + c$ form, then find where the line crosses both axes.

Solution

$3x + 2y = 12 \Rightarrow 2y = -3x + 12 \Rightarrow y = -\frac{3}{2}x + 6$. So $m = -\frac{3}{2}$, $c = 6$.

y-intercept: $(0, 6)$. x-intercept: set $y = 0$: $0 = -\frac{3}{2}x + 6 \Rightarrow x = 4$. So $(4, 0)$.

Key Takeaways — $y = mx + c$
  1. $y = mx + c$ is the gradient-intercept form of a straight line.
  2. $m$ is the gradient (steepness and direction); $c$ is the y-intercept.
  3. To write the equation: substitute $m$ and $c$ into $y = mx + c$.
  4. To read $m$ and $c$: compare term by term — $m$ is the coefficient of $x$, $c$ is the constant (including sign).
  5. To rearrange: make $y$ the subject, then simplify.
  6. Use the equation to predict points or verify if a point lies on the line.
Top pitfalls
Don't swap $m$ and $c$. Always keep the sign with $c$ (in $y = 2x - 3$, $c = -3$ not $+3$). $y = 2x$ means $c = 0$.

Lesson Complete!

You can now write, read and rearrange $y = mx + c$. Next: use these skills to find the equation of a line from a graph.