Introduction to Linear Equation in Two Variables

This topic gives an overview of;

- Linear Equation
- Solution of a Linear Equation

In earlier classes, you have studied linear equations in one variable. Write down a linear equation in one variable. You may say that * x + 1 = 0, x + √2 = 0 and √2 y + √3 = 0 *are examples of linear equations in one variable. You also know that such equations have a

Let us first recall what you have studied so far. Consider the equation:**2x + 5 = 0**. Its solution, i.e., the root of the equation, is **-5/2.** This can be represented on the number line as shown below:

This can be represented on the number line as shown below:

While solving an equation, you must always keep the following points in mind: The solution of a linear equation is not affected when:

- The same number is added to (or subtracted from) both the sides of the equation.
- You multiply or divide both the sides of the equation by the same non-zero number.

Let us now consider the following situation: In a One-day International Cricket match between India and Sri Lanka played in Nagpur, two Indian batsmen together scored 176 runs. Express this information in the form of an equation.

Here, you can see that the score of neither of them is known, i.e., there are **two unknown quantities**. Let us use x and y to denote them. So, the number of runs scored by one of the batsmen is x, and the number of runs scored by the other is y. We know that * x + y = 176 *which is the required equation.

This is an example of a linear equation in two variables. It is customary to denote the variables in such equations by x and y, but other letters may also be used. Some examples of linear equations in two variables are:

*1.2s + 3t = 5, p + 4q = 7, πu + 5v = 9 and 3 = 2 x – 7y*

Note that you can put these equations in the form * 1.2s + 3t – 5 = 0, p + 4q – 7 = 0, πu + 5v – 9 = 0 and 2 x – 7y – 3 = 0,* respectively.

So, any equation which can be put in the form** ax + by + c = 0**, where a, b and c are real numbers, and a and b are not both zero, is called a **linear equation in two variables**. This means that you can think of many many such equations.

You have seen that every linear equation in one variable has a unique solution. What can you say about the solution of a linear equation involving two variables? As there are two variables in the equation, a solution means a pair of values, one for x and one for y which satisfy the given equation. Let us consider the equation **2x + 3y = 12**. Here, **x = 3** and **y = 2** is a solution because when you substitute x = 3 and y = 2 in the equation above, you find that,

*2x + 3y = (2 × 3) + (3 × 2) = 12*

This solution is written as an **ordered pair (3, 2)**, first writing the value for x and then the value for y. Similarly, **(0, 4)** is also a solution for the equation above.

On the other hand, *(1, 4)* is not a solution of * 2x + 3y = 12*, because on putting x = 1 and y = 4 we get 2x + 3y = 14, which is not 12. Note that (0, 4) is a solution but not

You have seen at least two solutions for *2x + 3y = 12,* i.e., * (3, 2)* and (

On solving this, you get . so is another solution of .

Similarly, choosing x = – 5, you find that the equation becomes** **–10 + 3y = 12.

This gives so is another solution of .

So there is no end to different solutions of a linear equation in two variables. That is, a ** linear equation in two variables has infinitely many solutions**.

Let us consider an example to find four different solutions of the equation** x + 2y = 6**.

By inspection,** x = 2, y = 2** is a solution because for

Note that an easy way of getting a solution is to take x = 0 and get the corresponding value of y. Similarly, we can put y = 0 and obtain the corresponding value of x.

- An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables.
- Here a and b are called coefficients of x and y respectively and c is called constant term.
- The equation is called linear because the equation is of the first degree.
- A solution is an ordered pair of real numbers which satisfies the equation.
- A linear equation in two variables has infinitely many solutions.

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