Linear Equations In Two Variables Definition. An equation that can be put in the form ax + by + c = 0, where a, b and c are real numbers and a, b not equal to zero is called a linear equation in two variables namely x and y.

**Linear Equation in Two Variables:**

A linear equation in two variables is an equation that can be written in the form "ax + by = c", where "a", "b", and "c" are constants and "x" and "y" are variables. The general form of a linear equation in two variables is:

ax + by = c

**Solution of Linear Equations:**

The solution of a linear equation in two variables is an ordered pair (x, y) that satisfies the equation. In other words, when we substitute the values of "x" and "y" in the equation, the equation becomes true.

**Linear Equation in Two Variables: Ex 4.1 NCERT Solutions:**

In the exercise 4.1 of NCERT Solutions for Class 9, you will find various linear equations in two variables and their solutions. The exercise helps you practice solving linear equations in two variables using different methods such as the substitution method, elimination method, and cross-multiplication method.

**Example of a Linear Equation in Two Variables:**

Let's consider an example equation from the exercise:

2x + 3y = 7

This equation represents a straight line in a coordinate plane, where "x" and "y" are the coordinates of points on the line. The equation implies that for every value of "x", there is a corresponding value of "y" that satisfies the equation.

**Solving the Linear Equation:**

To find the solution of the equation, we need to substitute different values of "x" and find the corresponding values of "y" that make the equation true. We can also graph the equation on a coordinate plane and find the points of intersection with the x-axis and y-axis.

For example, if we substitute "x = 1" in the equation 2x + 3y = 7, we get:

2(1) + 3y = 7
2 + 3y = 7
3y = 7 - 2
3y = 5
y = 5/3

So, the solution of the equation is (1, 5/3).

Similarly, we can substitute different values of "x" to find more solutions of the equation.

**Summary:**

In summary, a linear equation in two variables represents a straight line on a coordinate plane. The solution of a linear equation is an ordered pair (x, y) that satisfies the equation. The NCERT Solutions for Class 9, Exercise 4.1, provides practice problems and solutions for solving linear equations in two variables using different methods.