Unit 1: Straight lines and Functions — MTH 201 Business Mathematics I Study Notes

Difficulty: ★★★ | High-yield focus: Master set operations (Union, Intersection, Complement), Venn diagrams, and real number system properties. Key exam mastery: Solve complex multi-set survey word problems and prove formal algebraic set relations.

Part 1: Core Concepts (2 Marks)

Standard Equations of a Straight Line

Slope-Intercept Form:
$y = mx + c$
Where $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$ is the gradient (slope), and $c$ is the $y$ -intercept.
** (★)Point-Slope Form:**
$y - y_1 = m(x - x_1)$
Two-Point Form:
$y - y_1 = \left(\frac{y_2 - y_1}{x_2 - x_1}\right)(x - x_1)$

Linear Economic Functions Formulation

Linear Demand Function: $Q_d = a - bP \quad (a > 0, b > 0)$
Linear Supply Function: $Q_s = -c + dP \quad (c \ge 0, d > 0)$
Linear Total Cost Function: $TC = FC + VC \cdot Q = F + vQ$
Linear Total Revenue Function: $TR = P \cdot Q$

The Method of Least Squares (Normal Equations)

To fit a straight line $y = a + bx$ to $n$ pairs of observations, we minimize the sum of squared errors $\sum e_i^2 = \sum (y_i - a - bx_i)^2$ . Setting partial derivatives to zero yields the Two Normal Equations:

\sum y = na + b\sum x

\sum xy = a\sum x + b\sum x^2

Solving simultaneously:

b = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}, \quad a = \bar{y} - b\bar{x}

Part 2: Structural Frameworks (10 Marks)

1. Geometric Properties of Linear Slopes

Economic Equilibrium & Coordinate Curve Dynamics

\text{Market Clearing Criterion: Quantity Demanded } = \text{ Quantity Supplied}

Economic State / Output Region	Curve Trajectory & Relationships	Analytical & Decision Rule
Market Equilibrium (E)	Demand curve intersects Supply curve ( $Q_d = Q_s$ )	Stable market price cleared with zero surplus or shortage
Price Above Equilibrium	Quantity Supplied exceeds Quantity Demanded ( $Q_s > Q_d$ )	Surplus creates downward competitive pressure on price
Price Below Equilibrium	Quantity Demanded exceeds Quantity Supplied ( $Q_d > Q_s$ )	Shortage creates upward competitive pressure on price

Parallel Lines Condition: Two lines with slopes $m_1$ and $m_2$ are parallel if and only if:
$m_1 = m_2$
Perpendicular Lines Condition: Two lines are perpendicular if and only if:
$m_1 \cdot m_2 = -1 \iff m_2 = -\frac{1}{m_1}$

2. Time Series Trend Forecasting via Least Squares Method

When forecasting annual business trends over $n$ years, coding the time variable $t$ such that $\sum t = 0$ dramatically simplifies computation:

Odd Number of Years ( $n = 5$ ): Set the middle year as origin $t = 0$ . Deviations: $-2, -1, 0, +1, +2$ . Since $\sum t = 0$ , the normal equations decouple instantly:
$a = \frac{\sum y}{n}, \quad b = \frac{\sum ty}{\sum t^2}$

Part 3: Distinction Tables

Comparison: Demand Curve vs. Supply Curve Geometry

Basis of Comparison	Linear Demand Curve	Linear Supply Curve
Slope Sign	Strictly negative ( $m < 0$ ); downward sloping from left to right.	Strictly positive ( $m > 0$ ); upward sloping from left to right.
Economic Rationale	Inverse relationship between price and quantity demanded (Law of Demand).	Direct relationship between price and quantity offered for sale (Law of Supply).
Intercepts	Positive $P$ -intercept (choke price) and positive $Q$ -intercept.	Positive $P$ -intercept (minimum reservation price) or negative $Q$ -intercept.

Part 4: Worked Problems & Exam Traps

Numerical Archetype: Fitting Line of Best Fit & Demand Forecasting

Problem: A commercial department store in Lalitpur records annual sales revenue ( $y$ , in NPR Lakhs) over a 5-year period:

Year	2020	2021	2022	2023	2024
Sales ( $y$ )	12	15	19	22	27

Fit a linear trend line $y = a + bt$ using the method of least squares.
Forecast expected sales revenue for the year 2026.

Step-by-Step Solution:

Step 1: Setup the Symmetry Deviation Table ( $n = 5$ ) Let the origin be the middle year $2022$ . Let $t = \text{Year} - 2022$ .

Year	Sales ( $y$ )	$t$	$t^2$	$t \cdot y$
2020	12	$-2$	4	$-24$
2021	15	$-1$	1	$-15$
2022	19	0	0	0
2023	22	$+1$	1	$+22$
2024	27	$+2$	4	$+54$
Total ( $n = 5$ )	$\sum y = 95$	$\sum t = 0$	$\sum t^2 = 10$	$\sum ty = 37$

Step 2: Calculate Parameters $a$ and $b$ Since $\sum t = 0$ :

a = \frac{\sum y}{n} = \frac{95}{5} = 19

b = \frac{\sum ty}{\sum t^2} = \frac{37}{10} = 3.7

Step 3: State the Fitted Trend Line Equation

y = 19 + 3.7t \quad (\text{Origin: 2022}, t \text{ unit: 1 year})

Step 4: Forecast for the Year 2026 For the year 2026, $t = 2026 - 2022 = +4$ :

y_{2026} = 19 + 3.7(4) = 19 + 14.8 = \text{NPR } 33.8 \text{ Lakhs}

Common Board Exam Trap: When given an even number of years (e.g., $n = 6$ ), setting $t = \text{Year} - \bar{X}$ produces half-integers ( $-2.5, -1.5, \dots$ ). To eliminate fractions, define $t = \frac{\text{Year} - \text{Midpoint}}{0.5}$ , producing integers ( $-5, -3, -1, +1, +3, +5$ ). In this case, remember that for a future year, the unit step is 2, not 1.

Students lose marks here because they make sign errors when applying De Morgan’s laws and calculating Cartesian set product subsets.

Business Mathematics I

Chapter notes

Unit 1: Straight lines and Functions· 6 LHs· ~8 marks