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LINEAR EQUATIONS IN TWO VARIABLES

A linear equation in one variable, like `x + 1 = 0, x + sqrt 2 = 0` and `sqrt 2 y + sqrt 3 = 0` are examples of linear equations in one variable.

You also know that such equations have a unique (i.e., one and only one) solution.

Here in this chapter, the knowledge of linear equations in one variable shall be recalled and extended to that of two variables.

You also know that such equations have a unique (i.e., one and only one) solution.

Here in this chapter, the knowledge of linear equations in one variable shall be recalled and extended to that of two variables.

Let us first recall what you have studied so far. Consider the following 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:

While solving an equation, you must always keep the following points in mind:

The solution of a linear equation is not affected when:

(i) the same number is added to (or subtracted from) both the sides of the equation.

(ii) 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:

`1.2 s + 3t = 5, p + 4q = 7, π u + 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 `sqrt 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.

`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:

While solving an equation, you must always keep the following points in mind:

The solution of a linear equation is not affected when:

(i) the same number is added to (or subtracted from) both the sides of the equation.

(ii) 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:

`1.2 s + 3t = 5, p + 4q = 7, π u + 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 `sqrt 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.

Q 3250267114

Write each of the following equations in the form `ax + by + c = 0` and indicate the values of a, b and c in each case:

(i) `2x + 3y = 4.37` (ii) `x – 4 = 3 y` (iii) ` 4 = 5x – 3y` (iv) `2x = y`

Class 9 Chapter 4 Example 1

(i) `2x + 3y = 4.37` (ii) `x – 4 = 3 y` (iii) ` 4 = 5x – 3y` (iv) `2x = y`

Class 9 Chapter 4 Example 1

(i) `2x + 3y = 4.37` can be written as `2x + 3y – 4.37 = 0`. Here `a = 2, b = 3`

and `c = – 4.37`.

(ii) The equation `x – 4 = sqrt 3 y` can be written as `x – 3 y – 4 = 0`. Here `a = 1`,

`b = – 3` and `c = – 4`.

(iii) The equation `4 = 5x – 3y` can be written as `5x – 3y – 4 = 0`. Here `a = 5, b = –3`

and `c = – 4`. Do you agree that it can also be written as `–5x + 3y + 4 = 0` ? In this

case `a = –5, b = 3` and `c = 4`.

(iv) The equation `2x = y` can be written as `2x – y + 0 = 0`. Here `a = 2, b = –1` and

`c = 0`.

Equations of the type `ax + b = 0` are also examples of linear equations in two variables

because they can be expressed as

`ax + 0.y + b = 0`

For example, `4 – 3x = 0` can be written as `–3x + 0.y + 4 = 0.`

Q 3260267115

Write each of the following as an equation in two variables:

(i) `x = –5` (ii) `y = 2` (iii) `2x = 3` (iv) `5y = 2`

Class 9 Chapter 4 Example 2

(i) `x = –5` (ii) `y = 2` (iii) `2x = 3` (iv) `5y = 2`

Class 9 Chapter 4 Example 2

(i) `x = –5` can be written as `1.x + 0.y = –5`, or `1.x + 0.y + 5 = 0`.

(ii) `y = 2` can be written as `0.x + 1.y = 2,` or `0.x + 1.y – 2 = 0`.

(iii) `2x = 3` can be written as `2x + 0.y – 3 = 0`.

(iv) `5y = 2` can be written as `0.x + 5y – 2 = 0`.