Chapter Overview
🧭Your visual roadmap through the chapter
The Coordinate System
Axes, quadrants, plotting points, slope, forms of a straight line
Application in Business
Cost, revenue, demand/supply and profit functions modelled as straight lines
Break-even Analysis
Contribution margin and break-even point, algebraically & graphically
Representing points and straight lines on an x–y coordinate plane, deriving the equation of a line in several forms, and reading where two lines intersect.
Straight-line modelling underlies cost/revenue/profit functions and break-even analysis — core techniques reused throughout financial mathematics and decision-making topics.
Plot and read coordinates; derive a line's equation from slope/points/intercepts; solve simultaneous equations graphically; compute break-even point and revenue.
Key Definitions
📘Source-based terminology, exam-ready
Coordinate System
A method of representing points in a space of given dimensions by coordinates; positions points in a two- or three-dimensional plane.
Key word: representing points
Coordinate
A group of numbers used to indicate the position of a point or line; by convention the x value is always given first, e.g. (10, 15).
Key word: (x, y) pair
Axes & Origin
The x axis (horizontal) and y axis (vertical) meet at right angles at the point of origin, usually denoted 0.
Key word: origin = 0
Quadrants
The origin divides the plane into 4 quadrants: Q1 (x,y both +), Q2 (x −, y +), Q3 (x,y both −), Q4 (x +, y −).
Key word: four regions
Slope (Gradient)
A number describing how steep a line is — the change in the dependent variable (y) resulting from a change in the independent variable (x).
Key word: steepness
y-Intercept / x-Intercept
y-intercept: value of y where the line crosses the y axis (x=0). x-intercept: value of x where the line crosses the x axis (y=0).
Key word: where it crosses
Plotting
Identifying the exact location of a point on the coordinate plane using its x and y values.
Key word: locating a point
Break-even Point
The sales volume at which total revenue equals total cost, i.e. no profit or loss — found where fixed cost is fully covered by contribution margin.
Key word: no profit, no loss
Core Concept
🎯Explained in exam-oriented language
CONCEPT → The Coordinate Plane & Straight Lines
MEANING
Any point on the plane is located by an (x, y) pair; a straight line is the set of all points satisfying a linear equation.
PURPOSE
Visualise and calculate relationships between two variables that change at a constant rate.
APPLICATION
Deriving a line's equation from slope, points or intercepts; reading where two lines intersect.
CONCEPT → Linear Functions in Business
MEANING
Cost, revenue and profit can be modelled as straight-line (linear) functions of the number of units.
PURPOSE
Find the point where total revenue equals total cost — the break-even point.
APPLICATION
Cost/average-cost function, revenue function, demand & supply function, profit/loss function, break-even (algebraic & graphical).
Key Rules & Principles
📏Every source rule, exam-ready
RULE 1 Quadrant sign rules
Quadrant 1: x and y both positive
Quadrant 2: x negative, y positive
Quadrant 3: x and y both negative
Quadrant 4: x positive, y negative
RULE 3 Intercept form — limitation
Only applies if the line intercepts BOTH the x and y axis
Does NOT apply to a line parallel to either axis
Does NOT apply to a line passing through the origin
RULE 4 Simultaneous equations on a graph
Two simultaneous equations can be plotted on a single graph
Where the two lines cross, there is a single (x, y) that satisfies both equations
Solving graphically = finding the point of intersection
RULE 2 Slope sign rules
Positive slope: upward left→right — x increase causes y increase
Negative slope: downward left→right — x increase causes y decrease
Zero slope: flat line (parallel to x-axis)
Infinite/undefined slope: vertical line (parallel to y-axis)
Formula Zone
🧮Every formula supported by the source text
SLOPE OF A STRAIGHT LINE
(x1, y1) and (x2, y2) are any two pairs of coordinates on the line.
SLOPE-INTERCEPT FORM
b = y-intercept (value where line crosses y axis, x=0) m = slope/gradient.
POINT-SLOPE FORM
(x1, y1) = a known point on the line; m = slope. Used when one point & the slope are known.
INTERCEPT FORM
a = x-axis intercept, b = y-axis intercept. Does not apply to lines through the origin or parallel to an axis.
TWO-POINT FORM
Used when two known points (x1,y1) and (x2,y2) are given.
BREAK-EVEN POINT & CONTRIBUTION MARGIN
FC=fixed cost, CM=contribution margin per unit, s=selling price/unit, v=variable cost/unit.
Calculation Roadmap
🗺️General path for any coordinate-geometry question
1· Read Question
Identify what's given: points, slope, intercepts
2· Identify Required
Equation, intercept, intersection or break-even?
3· Select Formula
Point-slope, two-point, intercept or BE formula
4· Substitute
Put known values into the formula
5· Calculate & Check
Solve, recheck, state final answer
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Equation from a Point + Slope (Point-Slope Form)
- 1Note the known point (x₁, y₁) and slope m
- 2Substitute into y − y₁ = m(x − x₁)
- 3Simplify to isolate y
- 4To find the y-intercept, set x = 0 and solve for y
METHOD C · Deriving the Line from Two Intercepts
- 1Note the x-intercept (a) and y-intercept (b)
- 2Substitute into x⁄a + y⁄b = 1
- 3Multiply through by (a × b) to clear fractions
- 4Rearrange into slope-intercept form y = b + mx
METHOD B · Equation from Two Points (Two-Point Form)
- 1Label the two given points (x₁,y₁) and (x₂,y₂)
- 2Substitute into (y−y₁)⁄(y₂−y₁) = (x−x₁)⁄(x₂−x₁)
- 3Cross-multiply and simplify
- 4Rearrange into the required form
METHOD D · Solving Simultaneous Equations Graphically
- 1Compute a table of (x, y) values for each equation
- 2Plot both lines on the same coordinate plane
- 3Identify the point where the two lines cross
- 4That (x, y) is the solution satisfying both equations
METHOD E · Break-even Point (Algebraic)
- 1Identify selling price (s), variable cost (v), fixed cost (FC)
- 2Calculate contribution margin per unit: CM = s − v
- 3Divide fixed cost by CM: BE (units) = FC ÷ CM
- 4For break-even revenue: BE units × selling price
Different Cases / Scenarios
🔀Recognise what's given, pick the right formula
IF one point and the slope are given
→ USE point-slope form: y−y₁=m(x−x₁)
IF two points on the line are given
→ USE two-point form
IF the x-intercept and y-intercept are given
→ USE intercept form: x/a + y/b = 1
IF the line passes through the origin, or is parallel to an axis
→ Intercept form does not apply — use slope-intercept or point-slope instead
IF two equations must be solved together
→ Plot both; the intersection point is the solution
IF selling price, variable cost & fixed cost are known
→ USE break-even formula BE = FC ÷ CM
IF a point's x and y are both positive
→ Point lies in Quadrant 1
IF x is negative and y is positive
→ Point lies in Quadrant 2
Comparison Tables
⚖️Don't confuse these!
| Form | Formula | When to use |
|---|---|---|
| Slope-Intercept | y = b + mx | Slope (m) and y-intercept (b) are known |
| Point-Slope | y−y₁=m(x−x₁) | One point and the slope are known |
| Intercept Form | x/a + y/b = 1 | Both the x-intercept (a) and y-intercept (b) are known |
| Two-Point Form | (y−y₁)/(y₂−y₁)=(x−x₁)/(x₂−x₁) | Two known points on the line are given |
| Quadrant | Sign of x | Sign of y |
|---|---|---|
| Quadrant 1 | Positive | Positive |
| Quadrant 2 | Negative | Positive |
| Quadrant 3 | Negative | Negative |
| Quadrant 4 | Positive | Negative |
Graph Interpretation
📈Reading coordinate-plane graphs
Reading a single line
The line crosses the y-axis at its y-intercept (b) and the x-axis at its x-intercept. Its steepness and direction (up or down, left to right) are read from the sign and size of the slope (m).
Reading two intersecting lines
When two equations are plotted together, the single point where the lines cross gives the (x, y) pair that satisfies both equations simultaneously — this is how simultaneous equations are solved graphically.
Worked Calculation Logic
✍️Method students must reproduce in the exam
GIVEN → FORMULA → SUBSTITUTION → CALCULATION → ANSWER
A company sells a product at Rs. 16,000/unit; variable cost Rs. 9,000/unit; fixed costs Rs. 50,000,000. Find the break-even sales volume and revenue.
GIVEN: s = 16,000, v = 9,000, FC = 50,000,000
FORMULA: CM = s − v then BE = FC ÷ CM
SUBSTITUTION: CM = 16,000 − 9,000 = 7,000 | BE = 50,000,000 ÷ 7,000
CALCULATION: BE = 7,142.857 units ≈ 7,143 units. Break-even revenue = 7,143 × 16,000
FINAL ANSWER: Break-even ≈ 7,143 units, or Rs. 114,288,000 in sales revenue
Exam Question Patterns
📝Slope/intercept — find slope, y-intercept or x-intercept from an equation
Equation-derivation — from two points, a point + slope, or two intercepts
Parallel/perpendicular lines — comparing slopes of two lines
Cost/total-cost function — deriving C(x) from two given cost points
Break-even application — units, revenue, or profit-function word problems
Exam Traps
⚠️⚠ WATCH OUT
- ✗ x-intercept vs y-intercept — mixing up which axis a value belongs to.
- ✗ Using intercept form on a line through the origin or parallel to an axis — it does not apply there.
- ✗ Quadrant sign errors — forgetting that Q2 and Q4 have one negative coordinate each.
- ✗ Sign slips substituting negative coordinates into point-slope or two-point form.
- ✗ Confusing CM per unit with total CM when applying the break-even formula.
Quick Decision Tree
🌳Chapter 2 in 60 Seconds
⚡Slope: (y₂−y₁)/(x₂−x₁)
Slope-intercept: y=b+mx
Point-slope: y−y₁=m(x−x₁)
Intercept form: x/a+y/b=1
Two-point form: ratio of differences
Break-even: BE=FC÷CM, CM=s−v Intersection = simultaneous solution
Trap: intercept form fails at origin
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