Overview
What you’ll learn
- Plot and read points in all four quadrants of the Cartesian plane.
- State the gradient and y-intercept of a line given in the form y = mx + c or ax + by = k.
- Draw the graph of a linear function from a table of values.
- Solve a pair of simultaneous linear equations graphically, by elimination and by substitution.
- Form and solve simultaneous equations from real-world problems.
In 30 seconds
- 1A point (x, y) is read across first, then up; the signs decide the quadrant.
- 2In y = mx + c, m is the gradient (how fast the line climbs) and c is the y-intercept (where it crosses the y-axis).
- 3Gradient = rise ÷ run between any two points on the line — it is the same everywhere on a straight line.
- 4Two lines meet at exactly one point unless they are parallel (same gradient, different intercepts) or the same line.
- 5The solution of a pair of simultaneous equations is the point that satisfies both — the intersection point of their graphs.
- 6Elimination adds or subtracts the equations to cancel one letter; substitution replaces one letter using the other equation.
Notes · 1 of 6
The Cartesian plane
Every point has an address: across first, then up.
The Cartesian plane is built from two number lines: the horizontal x-axis and the vertical y-axis, meeting at the origin (0, 0). A point is named (x, y): the x-coordinate says how far across, the y-coordinate says how far up. Negative coordinates move left or down.
| Quadrant | x | y | Example |
|---|---|---|---|
| I (top right) | + | + | (3, 2) |
| II (top left) | − | + | (−3, 2) |
| III (bottom left) | − | − | (−3, −2) |
| IV (bottom right) | + | − | (3, −2) |
Points on an axis belong to no quadrant: any point of the form (x, 0) sits on the x-axis, and (0, y) sits on the y-axis. Reading coordinates in the wrong order is the classic plot error — (2, 5) and (5, 2) are different points.
Notes · 2 of 6
Gradient and intercept: y = mx + c
Two numbers describe a straight line completely: its steepness and its start.
The gradient m measures steepness: gradient = rise ÷ run between any two points. A gradient of 3 means the line climbs 3 units for every 1 unit across; a negative gradient means the line falls as you move right.
| Equation | Gradient m | y-intercept c |
|---|---|---|
| y = 2x − 5 | 2 | −5 |
| y = −x + 4 | −1 | 4 |
| y = 7 | 0 (horizontal) | 7 |
| y = ½x | ½ | 0 (through the origin) |
Equations in the form ax + by = k hide the gradient and intercept until you rearrange: solve for y. From 2x + y = 6, subtract 2x: y = −2x + 6, so the gradient is −2 and the y-intercept is 6. The line crosses the x-axis where y = 0 — a different point that many questions ask about separately.
Notes · 3 of 6
Drawing a linear graph
Three honest points and a ruler beat any amount of guessing.
- 11. Make a table
Choose at least three x-values, including 0, and compute y = mx + c for each.
- 22. Plot the points
Mark each (x, y) pair carefully; negative coordinates go left or down.
- 33. Check for a straight line
The three points must line up — if not, one calculation is wrong.
- 44. Rule the line
Join them with a ruler and extend across the grid; label the line with its equation.
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | −1 | 1 | 3 | 5 | 7 |
The step pattern is worth noticing: each time x increases by 1, y increases by the gradient, 2 here. The table climbs −1, 1, 3, 5, 7 — always in steps of 2. If your table does not climb in equal steps, the equation has been misapplied.
Notes · 4 of 6
Solving simultaneous equations graphically
Two lines, one meeting point — that point solves both equations at once.
A simultaneous pair asks: which (x, y) makes both equations true? On a graph, that is exactly the intersection point of the two lines. Draw both lines on one pair of axes and read off where they cross.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y = x + 1 | 1 | 2 | 3 | 4 | 5 |
| y = −x + 5 | 5 | 4 | 3 | 2 | 1 |
Both rows show y = 3 when x = 2, so the lines cross at (2, 3) — the solution. You can check without drawing: put x = 2, y = 3 into each equation; both hold, so the point satisfies the pair.
Notes · 5 of 6
Elimination
Add or subtract the whole equations so one letter cancels.
Elimination works when one letter appears with matching coefficients. Subtracting kills matching terms; adding kills terms with opposite signs. x + y = 7 and x − y = 1: adding the two equations gives 2x = 8, because +y and −y cancel.
- 11. Line up the letters
Write both equations with x-terms, y-terms and constants in the same columns.
- 22. Cancel one letter
Add or subtract the full equations so one letter disappears.
- 33. Solve the survivor
The remaining one-letter equation gives one unknown immediately.
- 44. Back-substitute
Put that value into the easier original equation to find the other unknown.
Notes · 6 of 6
Substitution and real-world problems
When one equation is already solved for a letter, feed it into the other.
Substitution suits pairs where one equation names a letter directly, like y = x + 1. Replace y in the other equation: x + y = 9 becomes x + (x + 1) = 9, so 2x = 8, x = 4 — then y = 5. Use brackets when substituting, especially with negatives.
Word problems follow a fixed routine: give each unknown a letter, translate each sentence into one equation, then solve the pair by elimination or substitution. Money problems often need no drawing at all — the two receipts are the two equations.
Revise
Key terms
The words examiners expect you to use precisely.
- Cartesian plane
- The flat grid formed by a horizontal x-axis and a vertical y-axis.
- Origin
- The point (0, 0) where the two axes cross.
- Coordinates
- The pair (x, y) naming a point: across first, then up.
- Quadrant
- One of the four regions of the plane, decided by the signs of x and y.
- Gradient
- The steepness of a line: rise divided by run between any two of its points.
- y-intercept
- The point where a line crosses the y-axis; the value of c in y = mx + c.
Show 6 more key termsShow fewer key terms
- x-intercept
- The point where a line crosses the x-axis, found by putting y = 0.
- Linear function
- A function whose graph is a straight line, written y = mx + c or ax + by = k.
- Simultaneous equations
- A pair of equations that must both be true for the same values of the unknowns.
- Intersection point
- The point where two lines cross; it solves both equations at once.
- Elimination
- Solving a pair by adding or subtracting the equations to cancel one unknown.
- Substitution
- Solving a pair by replacing one unknown in an equation with its expression from the other.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Plotting (2, 5) as 5 across and 2 up.
Coordinates run x first, then y — across, then up.
Instead: (2, 5): 2 across, 5 up. Swapping the order lands on a different point.
Mistake: Reading the gradient of y = 5 − 2x as 5.
The gradient is the number attached to x, including its sign.
Instead: Rewrite as y = −2x + 5: gradient −2, intercept 5.
Show 5 more mistakesShow fewer mistakes
Mistake: Using the x-intercept as the value of c.
c is the y-intercept, where x = 0 — not where the line meets the x-axis.
Instead: Put x = 0 for c; put y = 0 for the x-intercept. They are usually different points.
Mistake: Adding only the left sides of an elimination pair.
The equals sign must be added too, or the new equation is meaningless.
Instead: Add left side to left side and right side to right side: (x + y) + (x − y) = 7 + 1.
Mistake: Substituting without brackets: putting y = x + 1 into x + y = 9 as x + x + 1.
Missing brackets usually survive the algebra by luck here, but with negatives like y = 2 − x they cause sign errors.
Instead: Substitute in brackets: x + (2 − x); the brackets cost nothing and protect the signs.
Mistake: Forgetting to find the second unknown.
A simultaneous pair needs both values; the question asks for a point, not a single letter.
Instead: Back-substitute immediately and finish with a stated pair (x, y), then check it in the unused equation.
Mistake: Declaring two same-gradient lines have a solution.
Same gradient and different intercepts means parallel — the lines never meet.
Instead: No intersection means no solution; identical lines mean infinitely many.
Check yourself
Quick quiz
10 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Locate points in the quadrants
The point (−2, 3) lies in which quadrant?
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 3Foundation
- 4Standard
- 3Challenging
10 questions worth 43 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 1 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 1.
Mark scheme
PROChapter 1 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 1Coming soonWeighted Assessment 1 (WA1)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 2 Math Chapter 1 about?
Chapter 1 covers linear graphs and simultaneous linear equations: the Cartesian plane, the gradient–intercept form y = mx + c, drawing straight-line graphs, and solving pairs of linear equations graphically, by elimination and by substitution.
What does y = mx + c tell me?
The m is the gradient — how much y rises for each step of 1 in x — and c is the y-intercept, where the line crosses the y-axis. For example, y = 2x − 5 climbs 2 units per step and starts at (0, −5).
When should I use elimination instead of substitution?
Use elimination when both equations are in the ax + by = k form and a letter can be cancelled by adding or subtracting (after multiplying if needed). Use substitution when one equation is already solved for a letter, such as y = x + 2.
What does the intersection of two lines mean?
It is the only point whose coordinates satisfy both equations, so it is the solution of the simultaneous pair. Parallel lines never intersect, so they have no solution; identical lines share every point and have infinitely many.
Up next
Chapter 2: Linear Inequalities