Q.A company produces shoes. When 30 shoes are produced the total cost of production is Rs. 1500. When 50 shoes are produced the costs increases to Rs. 2000. What is the cost equation (C) if it varies linearly in function to the number of shoes produced (q).

The Intuition: What Is a Straight Line?

Before any formula, think about what a straight line does. If you walk from your home to school, the shortest path is a straight line. If you stretch a thread tight between two pins, it forms a straight line. The key property is no turning — the direction never changes.

In coordinate geometry, we describe this mathematically. A straight line is the set of all points $(x, y)$ that satisfy a simple rule: the slope (steepness) between any two points on the line is constant.

The Precise Definition

A straight line in the $xy$-plane is the graph of a linear equation in two variables. The most general form is:

where $A$, $B$, and $C$ are real numbers, and $A$ and $B$ are not both zero.

The Slope: Why It Matters

The slope $m$ measures how steep the line is. It is the ratio of the vertical change to the horizontal change between any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line:

If the line is horizontal, $m = 0$. If it is vertical, the slope is undefined (division by zero). A positive slope means the line rises as you move right; a negative slope means it falls.

Common Forms of the Equation

You will encounter these forms frequently. Each is useful in different situations.

1. Slope-Intercept Form

Here $m$ is the slope and $c$ is the $y$-intercept — the point where the line crosses the $y$-axis. This is the most intuitive form: you see the slope and the starting height immediately.

2. Point-Slope Form

Use this when you know the slope $m$ and one point $(x_1, y_1)$ on the line.

3. Two-Point Form

Use this when you know two distinct points on the line.

4. Intercept Form

Here $a$ is the $x$-intercept and $b$ is the $y$-intercept. This form is convenient when the line cuts the axes at known points.

Special Cases to Watch For

• Line through the origin: $y = mx$ (no constant term).
• Horizontal line: $y = k$ (slope $0$).
• Vertical line: $x = k$ (slope undefined). …