Linear Equations in Two Variables — Class 9 IMO

Equations and Their Solutions

What is a Linear Equation in Two Variables?

A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are NOT both zero. The variables x and y have exponent 1 (they are linear).

Standard Form: ax + by + c = 0

Examples:

2x + 3y - 6 = 0 (a=2, b=3, c=-6)
x - y = 5 (can be rewritten as x - y - 5 = 0)
y = 2x + 1 (can be rewritten as 2x - y + 1 = 0)

Key Properties:

A linear equation in two variables has infinitely many solutions
Each solution is an ordered pair (x, y) that satisfies the equation
When plotted on a coordinate plane, all solutions lie on a straight line

Why "Linear"?

The word "linear" comes from the Latin word "linea" meaning line. The graph of such an equation is always a straight line.

Real-Life Applications:

Relationship between cost and quantity (C = mx + b)
Distance-time relationship (d = rt)
Budget constraints (2x + 3y = 100)

Example 1: Is (2, 1) a solution of x + y = 3?

2 + 1 = 3, so yes, (2, 1) satisfies it.

Example 2: Give any solution of 2x + y = 5.

Take x = 1 ⇒ y = 3, so (1, 3) is one of infinitely many.

Quick recap

Form: ax + by + c = 0, a and b not both 0.
A solution is a pair (x, y); there are infinitely many.

✓ Quick check

A standard linear equation ax + by + c = 0 reduces to a line parallel to the x-axis if:

A line parallel to the x-axis is of the form y = k. For ax + by + c = 0 to reduce to this form, the x term must vanish, which requires coefficient a = 0 while b must be non-zero.

The positive solutions of the equation ax + by + c = 0 always lie in the:

Positive solutions mean both x > 0 and y > 0. In the coordinate plane, both coordinates are positive only in the 1st quadrant.

Graph of a Linear Equation

What is the Graph of a Linear Equation?

The graph of a linear equation in two variables is the set of all points (x, y) whose coordinates satisfy the equation. This graph is always a straight line.

Steps to Graph a Linear Equation:

Find at least three solutions (ordered pairs) of the equation
Plot these points on the Cartesian plane
Draw a straight line passing through all plotted points
Extend the line in both directions with arrows

Why Three Points?

Two points determine a line, but using a third point helps verify accuracy — all three should be collinear (lie on the same line).

Graphing Special Cases:

Equation Type	Example	Graph Description
y = constant	y = -2	Horizontal line through (0, -2)
Through origin	y = 2x	Line passing through (0, 0)
ax + by = 0	2x + y = 0	Line through origin

Intercepts Method:

x-intercept: Point where line crosses x-axis (y = 0)
y-intercept: Point where line crosses y-axis (x = 0)

Example 1: Two points on y = x + 1?

x = 0 ⇒ (0, 1); x = 2 ⇒ (2, 3). Join them for the line.

Example 2: Does the graph of x + y = 4 pass through (4, 0)?

4 + 0 = 4, so yes — it is on the line.

Quick recap

The graph of a linear equation is a straight line.
Plot two or three solution points, then join with a ruler.

✓ Quick check

Find the value of k if x = 3k + 2 and y = 2k − 1 is a solution of the equation 4x − 3y + 1 = 0.

Substitute x and y into 4x − 3y + 1 = 0: 4(3k + 2) − 3(2k − 1) + 1 = 0 → 12k + 8 − 6k + 3 + 1 = 0 → 6k + 12 = 0 → 6k = −12 → k = −2.

Find the value of p if x = p and y = 2 is a solution of the equation 3x − 4y = 7.

Substitute x = p and y = 2 into 3x − 4y = 7: 3(p) − 4(2) = 7 → 3p − 8 = 7 → 3p = 15 → p = 5.

Lines Parallel to the Axes

An equation with only one variable still draws a line in the plane. x = k is a vertical line through (k, 0), parallel to the y-axis; y = k is a horizontal line through (0, k), parallel to the x-axis.

So x = 0 is the y-axis itself and y = 0 is the x-axis. Word problems often translate a relationship into such an equation, then a graph.

Example 1: What does x = 3 look like?

A vertical line crossing the x-axis at 3, parallel to the y-axis.

Example 2: Which line is y = 0?

It is the x-axis.

Quick recap

x = k is vertical; y = k is horizontal.
x = 0 is the y-axis; y = 0 is the x-axis.

✓ Quick check

In an auto-rickshaw fare system in New Delhi, the fare for the first kilometre is ₹20 and for subsequent distance it is ₹12 per km. If the total distance covered is x km and total fare is ₹y, write the linear equation.

Total distance = x km. Distance after 1st km = (x − 1) km. Fare for 1st km = ₹20. Fare for remaining distance = 12(x − 1). Total fare y = 20 + 12(x − 1) = 20 + 12x − 12 = 12x + 8.

A structural pillar is constructed using pre-cast segments. The total height y (in meters) after adding x segments is given by y = 2.5x + 1.5. If the total height reached is 14 meters, how many segments were used?

Substitute y = 14 into the equation: 14 = 2.5x + 1.5 → 14 − 1.5 = 2.5x → 12.5 = 2.5x → x = 12.5 / 2.5 = 5 segments.