What is Linear Equations in Two Variables Class 9: Definition & Examples

Definition of Linear Equations in Two Variables

A linear equation in two variables is an algebraic equation of the form:

$$ax + by + c = 0$$

where:

• $x$ and $y$ are variables,
• $a$, $b$, and $c$ are constants with $a \neq 0$ and $b \neq 0$.

This means the highest power of both variables is 1. The equation represents a straight line when plotted on the Cartesian plane. For example, $2x + 3y - 6 = 0$ is a linear equation in two variables.

In Class 9 NCERT Mathematics, understanding this definition helps students identify and work with such equations effectively.

General Form and Characteristics

The general form of a linear equation in two variables is:

$$ax + by + c = 0$$

Key characteristics include:

• Both $x$ and $y$ appear to the power 1 only.
• No product of variables (like $xy$) or higher powers (like $x^2$) are present.
• The equation can be rewritten as $y = mx + c'$ where $m$ is the slope.

This form is used extensively in Class 9 NCERT problems to solve and graph linear equations.

How to Find Solutions of Linear Equations in Two Variables

A solution of a linear equation in two variables is an ordered pair $(x, y)$ that satisfies the equation.

Example: Consider the equation

$$2x + 3y = 6$$

To find solutions, assign a value to one variable and solve for the other:

• If $x=0$, then $3y=6 \Rightarrow y=2$; solution: $(0,2)$
• If $x=3$, then $2(3)+3y=6 \Rightarrow 6 + 3y=6 \Rightarrow 3y=0 \Rightarrow y=0$; solution: $(3,0)$

You can find infinitely many solutions this way. Plotting these points will form a straight line representing all solutions.

Graphical Representation of Linear Equations

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

Steps to plot:

1. Find at least two solutions (points) of the equation. 2. Plot these points on the Cartesian plane. 3. Draw a straight line passing through these points.

Example: For $x + y = 4$,

• If $x=0$, then $y=4$; point $(0,4)$
• If $y=0$, then $x=4$; point $(4,0)$

Plotting these points and joining them gives the graph of the equation.

This visual understanding is emphasized in Class 9 NCERT to connect algebra and geometry.

Methods to Solve Linear Equations in Two Variables

In Class 9, solving linear equations in two variables often involves finding values of $x$ and $y$ that satisfy two such equations simultaneously.

Common methods include:

• Substitution Method: Solve one equation for one variable and substitute into the other.
• Elimination Method: Add or subtract equations to eliminate one variable.

Example: Solve

$$x + y = 5$$ $$2x - y = 4$$

Using substitution: From first, $y=5 - x$. Substitute in second:

$$2x - (5 - x) = 4 \Rightarrow 2x - 5 + x = 4 \Rightarrow 3x = 9 \Rightarrow x = 3$$

Then,

$$y = 5 - 3 = 2$$

Solution: $(3, 2)$

Mastering these methods is vital for Class 9 NCERT exam success.

Real-Life Applications of Linear Equations in Two Variables

Linear equations in two variables are not just theoretical; they model many real-life situations such as:

• Calculating cost and quantity in business.
• Determining distance and speed relationships.
• Budget planning and resource allocation.

Example: If a taxi charges ₹50 as base fare plus ₹10 per km, the total cost $C$ for $x$ km is:

$$C = 10x + 50$$

Here, $C$ and $x$ are variables, and the equation is linear in two variables.

Understanding these applications helps Class 9 students see the importance of this chapter beyond exams.