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 equation that can be written in the form:

$$ax + by + c = 0$$

where $x$ and $y$ are variables, and $a$, $b$, and $c$ are constants with $a \neq 0$ and $b \neq 0$. The degree of each variable is 1, meaning the variables are not raised to any power other than one.

For example, $2x + 3y - 5 = 0$ is a linear equation in two variables. Here, $a=2$, $b=3$, and $c=-5$.

This equation represents a straight line when plotted on the Cartesian plane.

General Form and Components of Linear Equations

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

$$ax + by + c = 0$$

• $a$ and $b$ are coefficients of variables $x$ and $y$ respectively.
• $c$ is a constant term.
• $x$ and $y$ are variables representing unknown values.

Important points:

• Both $a$ and $b$ cannot be zero simultaneously.
• If $a=0$, the equation reduces to $by + c = 0$ which is a linear equation in one variable $y$.
• Similarly, if $b=0$, it reduces to $ax + c = 0$.

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.

Steps to find solutions:

1. Choose any value for one variable (say $x$). 2. Substitute this value into the equation. 3. Solve for the other variable ($y$).

Example:

Find two solutions of the equation $2x + 3y = 6$.

• Let $x = 0$:

$$2(0) + 3y = 6 \Rightarrow 3y = 6 \Rightarrow y = 2$$ Solution: $(0, 2)$

• Let $x = 3$:

$$2(3) + 3y = 6 \Rightarrow 6 + 3y = 6 \Rightarrow 3y = 0 \Rightarrow y = 0$$ Solution: $(3, 0)$

Any such pair that satisfies the equation is a solution.

Graphical Representation of Linear Equations

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

Plotting steps:

• Find at least two solutions (points) of the equation.
• Plot these points on the Cartesian plane.
• Join the points with a straight line.

Example:

Graph the equation $x + y = 4$.

• Let $x=0$, then $y=4$ → Point $(0,4)$
• Let $y=0$, then $x=4$ → Point $(4,0)$

Plot these points and draw a line through them. This line represents all solutions of the equation.

Note: Every point on this line is a solution to the equation.

Applications of Linear Equations in Two Variables

Linear equations in two variables are used to solve many real-life problems involving two quantities.

Examples include:

• Calculating cost and quantity in shopping.
• Distance and speed problems.
• Mixing solutions or ingredients in given ratios.

Example problem:

A shop sells pens and pencils. A pen costs ₹5 and a pencil costs ₹3. If a customer buys 4 items and spends ₹18, find how many pens and pencils were bought.

Let $x$ = number of pens, $y$ = number of pencils.

Equations:

$$x + y = 4$$ $$5x + 3y = 18$$

Solving these simultaneously gives the quantities of pens and pencils.

This shows how linear equations help model and solve practical problems.

Difference Between Linear Equations in One and Two Variables

Understanding the difference is important for Class 9 students.

This comparison helps clarify concepts and exam preparation.