Chapter 4: Linear Equations

1. Standard Form

A linear equation in two variables can be written in the form:

\( ax + by + c = 0 \)

Where \( a, b, c \) are real numbers, and \( a \) and \( b \) are not both zero.

Feature	One Variable	Two Variables
Form	\( ax + b = 0 \)	\( ax + by + c = 0 \)
Solutions	Unique (One)	Infinitely Many
Graph	Point on Number Line	Straight Line on Cartesian Plane

Coefficient Hunter

Equation: \( 2x + 3y = 4.37 \)

Step 1: Write as \( 2x + 3y - 4.37 = 0 \)

Now identify a, b, and c:

2. Solutions

A linear equation in two variables has infinitely many solutions. A solution is a pair of values \( (x, y) \) that satisfies the equation.

How to find a solution?

Tabular Method (Very Important)

Let’s find solutions of \( x + y = 6 \)

x	y	Ordered Pair
0	6	(0, 6)
2	4	(2, 4)
4	2	(4, 2)

1. Pick any value for \( x \) (e.g., \( x = 0 \)).

2. Substitute it into the equation.

3. Solve for \( y \).

Example: For \( x + 2y = 6 \):

If \( x = 0 \), then \( 2y = 6 \Rightarrow y = 3 \). So \( (0, 3) \) is a solution.

🧠 Why infinitely many solutions?

Because one equation with two variables cannot fix both values. If x changes, y can adjust to still satisfy the equation.

CBSE Language: One equation → two unknowns → infinite solutions.

Solution Verifier

Find a solution for: \( 2x + y = 7 \)

3. Graphing Lines

The graph of every linear equation in two variables is a straight line.

Why is the graph always a straight line?

Every solution of a linear equation lies on the same straight path. When we plot many such solutions, they align to form a straight line.

Equation: \( x = 2 \)

Parallel to Y-axis

Equation: \( y = 3 \)

Parallel to X-axis

🚨 Common CBSE Mistakes

Thinking linear equation has only one solution ❌
Confusing x = a with y = a ❌
Forgetting to convert to standard form ❌
Not writing ordered pairs clearly ❌

Avoid these → easy marks saved 🎯

Key Points:

Equation of X-axis is \( y = 0 \).
Equation of Y-axis is \( x = 0 \).
\( x = a \) is a vertical line parallel to Y-axis.
\( y = a \) is a horizontal line parallel to X-axis.

🧩 Is This a Solution?

Check without calculating fully:

Is (2, 3) a solution of \( x + y = 5 \)?
Is (0, 4) a solution of \( 2x + y = 4 \)?
Is (1, 1) a solution of \( x = 1 \)?

Show Answers

Yes
Yes
Yes

🎯 Exam Smart Zone

Graph questions are always 3 marks
Two points are enough to draw a line
Axes equations are guaranteed MCQs
Always write solution as (x, y)

Tip: Neat graph = full marks.

Concept Mastery Quiz

1. How many solutions does \( 2x + 5y = 7 \) have?

A) One unique solution

B) Two solutions

C) Infinitely many solutions

2. The equation of the X-axis is:

A) \( x = 0 \)

B) \( y = 0 \)

C) \( x = y \)

3. If the point (1, -2) lies on \( 2x - y = k \), find k.

A) 4

B) 0

C) -3

4. The graph of \( x = 5 \) is a line:

A) Parallel to Y-axis

B) Parallel to X-axis

C) Passing through origin

5. Which of the following is a solution of \( x - 2y = 4 \)?

A) (0, 2)

B) (2, 0)

C) (4, 0)