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Pair of Linear Equations in Two Variables

Linear equations in two variables are equations where we have two variables. We require two equations to find the solution of linear equations in two variables. Go through the following video to understand the basic concept of linear equations in two variables.

Now the given video will help you to frame equations for a given word problem.

Remember: To solve a linear equation in one variable, only one equation is required. To solve linear equations in two variables, two linear equations in the same two variables are required. Likewise, we can say that to solve linear equations with n number of variables, n numbers of linear equations in the same n number of variables are required.

The general form of a pair of linear equations in two variables is written as

, where a1, b1 are real numbers and (i.e., a1, b1 ≠ 0)

, where a2, b2 are real numbers and (i.e., a2, b2 ≠ 0)

We know that the solution of a linear equation must satisfy the equation. Conversely, we can say that the value of the variable in the equation, which satisfies the equation, is the solution of the equation.

In case of a pair of linear equations, the values of x and y, which satisfy both the equations, are the solutions of the equation.

Geometrically, a linear equation represents a straight line. Every solution of an equation is a point on the line represented by the equation.

Likewise, a pair of linear equations represents two straight lines. When their graphs are drawn, there are three possibilities. Let us go through the video to understand the various possibilities.

Let us now consider some more examples to check whether the pair of equations is consistent, inconsistent, or dependent.

Example 1:

Find whether the given pairs of linear equations are consistent or inconsistent?

(a) 5x + 2y = 2

20x + 8y = 1

(b) 4x + y = 8

7x – 2y = 1

(c) 5x + 6y = 9

10x + 12y = 18

Solution:

(a) The solution is explained with the help of the following video. Let us go through the video to understand it well.

(b)4 x + y = 8

7x – 2y = 1

Here, a1 = 4 b1 = 1 c1 = –8
a2 = 7 b2 = –2 c2 = –1

Therefore, according to the above rule, this system of equations has a unique solution. It is represented by intersecting lines.

(c) 5x + 6y = 9

10x + 12y = 18

Here, a1 = 5 b1 = 6 c1 = –9
a2 = 10 b2 = 12 c2 = –18

Therefore, according to the above rule, this system of equations has infinite solutions. It is represented by coincident lines.

Example 2:

There are two lines drawn in each of the following graphs. Each line represents a linear equation in two variables. St...

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