Think First
warm-up

Before diving into the practice questions, take a moment to think about what you already know about this topic.

Record in workbook.
1
What You'll Master
objectives

Know

  • A linear equation $y = mx + c$ produces a straight-line graph

Understand

  • $m$ is the gradient — it controls the steepness and direction of the line
  • $c$ is the y-intercept — it shows where the line crosses the y-axis

Can Do

  • Complete a table of values for a linear equation
  • Plot and connect points to draw a straight-line graph
  • Read the gradient and y-intercept from the equation $y = mx + c$
2
Words You Need
vocabulary
Linear RelationshipA relationship between two quantities that forms a straight line when graphed. The rule can always be written as $y = mx + c$.
Gradient (Slope)How steep a line is. A positive gradient means the line rises left to right; a negative gradient means it falls. Calculated as rise ÷ run.
y-interceptThe point where the line crosses the y-axis. It happens when $x = 0$. In $y = mx + c$, the y-intercept is $c$.
Table of ValuesA table where you substitute chosen $x$ values into a rule to find the matching $y$ values. Gives you coordinates to plot.
$y = mx + c$The standard form of a linear equation. $m$ is the gradient, $c$ is the y-intercept, $x$ is the input, and $y$ is the output.
Rise over RunThe formula for gradient: $\text{gradient} = \dfrac{\text{rise}}{\text{run}} = \dfrac{\text{change in }y}{\text{change in }x}$. Rise = vertical change, run = horizontal change.
3
Table of Values Method
+5 XP to read

To graph a linear equation, substitute a set of $x$ values to find the matching $y$ values. Each pair $(x, y)$ is a point you can plot on the Cartesian plane.

For $y = 2x + 1$, substitute $x = -2, -1, 0, 1, 2$:

$x$ −2 −1 0 1 2
$y$ −3 −1 1 3 5

Plot each $(x, y)$ pair on the Cartesian plane, then connect them with a straight line.

x y −2 −1 0 1 2 (0,1)
$y = 2x + 1$
2 points minimum
You only need 2 points to draw a straight line — but use 3 or more to catch arithmetic mistakes.
Use $x = 0$ first
Starting with $x = 0$ gives you the y-intercept straight away — the easiest point to plot.
Points in a line?
If your 3 points don't line up, one calculation is wrong — go back and check each substitution.
4
Gradient and y-intercept from $y = mx + c$
+5 XP to read

Once you know the form $y = mx + c$, you can read off the gradient and y-intercept instantly — no table needed.

In $y = mx + c$: $m$ is the gradient and $c$ is the y-intercept.

For $y = 3x - 2$:

  • Gradient $= 3$ (rises 3 units for every 1 unit right)
  • y-intercept $= -2$ (line crosses y-axis at $-2$)

Positive gradient → line slopes up (left to right).
Negative gradient → line slopes down.

$$\text{Gradient} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in }y}{\text{change in }x}$$

rise run m = rise ÷ run
$y = \underbrace{3}_{m}x \underbrace{- 2}_{c}$
$m$ comes first
In $y = mx + c$, the number in front of $x$ is always $m$. In $y = 3x - 2$, it's 3 — not $-2$.
$c$ is the stand-alone number
$c$ is the number not multiplied by $x$. Watch signs: in $y = 4x - 7$, $c = -7$, not 7.
Steeper = larger $|m|$
A gradient of 5 is steeper than a gradient of 2. The bigger the absolute value, the steeper the line.
5
Real-World Linear Models
+5 XP to read

Linear equations appear everywhere in real life. Any situation with a fixed starting amount and a constant rate of change can be modelled by $y = mx + c$.

Scenario: A plumber charges an \$80 call-out fee plus \$60 per hour.

$C = 60h + 80$

  • Gradient $= 60$ → cost per extra hour
  • y-intercept $= 80$ → fixed call-out fee
Hours ($h$) 0 1 2 3 4
Cost ($C$) \$80 \$140 \$200 \$260 \$320
h C 80 1 2 3 4
$C = 60h + 80$
Gradient = rate of change
In real-world models, the gradient is always the rate: dollars per hour, metres per second, etc.
y-intercept = starting value
The y-intercept is the value when $x = 0$ — the fixed fee, starting balance, or initial amount.

Watch Me Solve It · Worked example

Watch Me Solve It · Graphing $y = -x + 3$
+15 XP per step
Q
PROBLEM
Graph $y = -x + 3$ for $x = 0, 1, 2, 3$. Identify the gradient, y-intercept, and x-intercept.
  1. 1
    Build the table of values
    $(0,\ 3),\quad (1,\ 2),\quad (2,\ 1),\quad (3,\ 0)$
    Substitute each $x$ into $y = -x + 3$. For $x = 0$: $y = 0 + 3 = 3$. For $x = 1$: $y = -1 + 3 = 2$. And so on.
  2. 2
    Read the gradient and y-intercept
    Gradient $= -1$  (the coefficient of $x$)    y-intercept $= 3$  (the constant)
    In $y = -1x + 3$, $m = -1$ and $c = 3$. A gradient of $-1$ means the line goes down 1 unit for every 1 unit to the right.
  3. 3
    Describe plotting the points
    Start at $(0, 3)$. Move 1 right and 1 down for each next point.
    Because the gradient is $-1$: run = 1, rise = $-1$. Plot $(0,3)$, then go right 1, down 1 to reach $(1,2)$, $(2,1)$, $(3,0)$. Connect with a ruler.
  4. 4
    Find the x-intercept
    Set $y = 0$: $\quad 0 = -x + 3 \quad\Rightarrow\quad x = 3$
    The x-intercept is where the line crosses the x-axis (when $y = 0$). Solving gives $x = 3$, so the line crosses at $(3, 0)$ — which matches our table.
Answer Gradient $= -1$ (line slopes down) · y-intercept $= 3$ · x-intercept $= 3$ at $(3,0)$
6
Common Pitfalls
heads-up
Swapping $m$ and $c$ in $y = mx + c$
In $y = mx + c$: $m$ is the gradient (number in front of $x$) and $c$ is the y-intercept (the stand-alone number) — not the other way around. In $y = 4x + 7$, the gradient is 4 and the y-intercept is 7.
Fix: ask "what's multiplying $x$?" That's $m$. Then ask "what's left by itself?" That's $c$.
Using only 2 points then making an error
A line only needs 2 points to define it, but if either calculation has an arithmetic error, you'll draw the wrong line. Using 3 or more points gives you a built-in check — if they're not collinear, you've made a mistake.
Fix: always calculate at least 3 points and verify they form a straight line before connecting them.
Calculating gradient as run ÷ rise
Gradient = rise ÷ run, not run ÷ rise. Rise is the vertical change ($y$), run is the horizontal change ($x$). Writing it upside-down gives the wrong gradient every time.
Fix: remember "rise over run" — rise (up/down, $y$) is on top; run (left/right, $x$) is on the bottom. $m = \dfrac{\Delta y}{\Delta x}$

Quick Check · 5 questions

1
Which table matches $y = 3x + 2$?
+10 XP
2
A plumber charges $40 callout plus $30 per hour. Which rule gives the cost $C$ for $h$ hours?
+10 XP
3
For $y = 2x + 5$, what is the y-intercept?
+10 XP
4
A graph passes through $(0, 3)$ and $(2, 7)$. What is its equation?
+10 XP
5
From the graph of $y = 3x + 1$, what is $y$ when $x = 5$?
+10 XP
Apply Easy 2 MARKS

Q1. Complete the table of values for $y = 4x - 2$, then describe the pattern in the $y$ values.

Answer in your workbook.
Apply Medium 3 MARKS

Q2. A gym membership costs $20 joining fee plus $12 per week. Write the rule and calculate the cost for 15 weeks.

Answer in your workbook.
Apply Hard 4 MARKS

Q3. A line passes through $(1, 5)$ and $(3, 11)$. Find the slope and y-intercept, and write the equation of the line.

Answer in your workbook.
Stretch Challenge · +25 XP, +10 coins

Extension Problems

A car rental costs $50 per day plus $0.30 per kilometre. Write a rule and find the cost for driving ...

R
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