Linear Equations

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 .

This can be represented on the graph 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:

1.2s+3t=5, p+4q=7, πu+5v=9 and 3=2x−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 2x−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.

Example 1

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

Solution

2x + 3y = 4.37 can be written as 2x + 3y – 4.37 = 0. Here a = , b = and c = .

(ii)

x−4=3y

Solution

The equation x−4=3y can be written as x−3y−4=0. Here a = , b = and c = .

(iii)

4 = 5x – 3y

Solution

The equation 4 = 5x – 3y can be written as 5x – 3y – 4 = 0. Here a = , b = and c = .

Do you agree that it can also be written as −5x+3y+4=0 ?

In this case, a = , b = and c = .

(iv)

2x = y

Solution

The equation 2x=y can be written as 2x−y+0=0. Here a = , b = and c = .

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 x + y + = 0.

Let's see the more examples.

Example 2

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

(i)

x=−5

Solution

x=−5 can be written as x + y + = 0.

(ii)

y=2

Solution

y=2 can be written as x + y – = 0.

(iii)

2x=3

Solution

2x=3 can be written as x + y – = 0.

(iv)

5y=2

Solution

5y=2 can be written as x + y – = 0.

Find the value of variables which satisfies the following equations:

8x + 7y = 38 and 3x - 5y = - 1.

Instructions

Find the value of variables

Using the method of substitution to solve the pair of linear equation, we have eq: 8x + 7y = 38 (i)
eq: 3x - 5y = -1 (ii)
Multiplying equation (i) by 5 and then we get: x + y = (iii)
Multiplying equation (ii) by 7, and then we get : x - y = (iv)
Adding equation (iv) and (iii), we get: x =
Therefore ⇒ x =
Substituting the value of x in either equation (i) or (ii), we have to apply the value in equation (ii) is
Therefore ⇒ y =
Therefore, x = 3 and y = 2 is the point where the given lines intersect.

After knowing how to solve a pair of linear equations algebraically, let us represent an equation on a coordinate plane. These equations tend to have infinitely many solutions.

The solutions of the linear equation: 3x + 2y = 6 can be illustrated in the form of a table as follows by writing the values of y below the corresponding values of x :

Plotting Graph of Linear Equations in Two Variables

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