Any equation that can be represented in the form of ax + by + c =o, where a,b, and c are real numbers and a, b are not equivalent to 0 is known as a linear equation in two variables namely x and y. The solutions for such types of equations are a pair of values, one for x and one for y which makes the two sides of the equation equal.
There are infinitely multiple solutions for linear equations in two variables. For example, x + 2y = 6 in a linear equation and its solutions are (0,3), (6,0), (2,2) because they satisfy the equation x + 2y = 6.
Here, we will understand the linear equation in two variables through an example
Let us take the equation
5x + 3y = 30
The above equation has two variables i.e. x and y
This equation can be represented graphically through substituting the variables equals zero.
The value of x, when y equals to zero is
5x + 3(0) = 30
→ x = 6
And, the value of y when x equals to zero is
5 (0) + 3y = 30
Y = 10
As the solutions of linear equations in two variables is a pair of numbers (x,y), we can express the solutions in a coordinate plane.
Let us understand the concept by considering the equation given below:
2x + y = 6 (1)
Some of the solution of the equations given above are:
(0,6), (3,0), (1,4) ,(2,2) because they satisfy the equation 2x + y = 6
Now, we are plotting the above coordinates in the coordinate plane below.
We can take any two of the coordinates and join them to form a line. Let that line be PQ. It can be seen in the below figure that all the 4 coordinates are lying on the same line PQ.
Let us take any other point such as ( 4,-2) which lies on the line PQ.
Now, we will verify whether the above point is satisfying the equation or not.
Substituting the point ( 4,-2) in equation (1) we get,
LHS = ( 2*4) -2 = 6 = RHS
Therefore, the coordinate (4,-2) is solution of 2x + y = 6
Similarly, if we will take any other points, it will also satisfy the equation 2x + y = 6
It can be seen that all the points lie on the line PQ provide a solution of 2x + y = 6
All the solution of 2x + y = 6 lies on the line PQ
Coordinates that will not satisfy the equation of 2x + y = 6 will not lie in the line PQ.
Important points of graphing of linear equation in two variables
We can conclude the following points, for a linear equation in two variables
Each point on the line will be the solution to the equation.
Each solution of the equation will be some point on the line
Hence, we can represent every linear equation in two variables in a graph as a straight line in a coordinate plane. Points on the lines are known as the solution of the equation. Due to this, an equation with one degree is known as linear equations. The expression of linear equations in a graph is known as graphing of linear equations in two variables.
1. 10 students of class 9th took part in a Science quiz. If the number of girls participated in a quiz is 4 times more than boys, find the number of girls and boys who took part in a Science quiz.
Solution: Let the number of boys participated by y and the number of girls participated by x.
Accordingly, equation will be
x + y = 10 (1)
y = x + 4 (2)
Let us now represent the above equations 1 and 2 graphically by calculating 2 solutions for each of the equations. The two solutions of the equations are:
x + y = 10 → y = 10 - x
x + y = 4
Now we will plot the above points in a graph,
We will draw two lines AB and CE passing through the points to represent the equation
The two lines AB and CE will intersect at point E ( 3, 7).
Hence, x = 3 and y = 7 is the required solution of the pair of linear equations.
So, the total number of boys participated in quiz = 3 and the total number of girls participated in the quiz = 7
Substituting the values of x =3 and y= 7 in the equation (1) , we get
L.H.S = 3 + 7= 10
LHS = RHS
Substituting the values of x =3 and y= 7 in the equation (2) , we get
7 = 3 + 4
LHS = RHS
2. Represent graphically that the following system of equation 2x + 3y = 10 and 4x + 6y = 2 has no solution.
Solution: The given equations are:
2x + 3y = 10 → y = (10 - 2x) /3
4x + 6y = 2 → y = (12 - 4x) /6
No, we are plotting the points A ( -4,6) and B (2,2) in a graph. Join these two points to form a line AB.
Also, plot the points C ( -3,4) and D ( 3,0) and join them to form a line CD
Now, you can see the lines in the graph are parallel to each other. As the line has no common points, there will be no common solution.
Therefore, the given system of the equation has no solution
1. The graph of x= -2 in a line parallel to the
Both x and y- axis
None of these
2. If the lines represented by 2x + ky = 1 and 3x - 5y = 7 ae parallel, then value of k is
3. What will be the representation of the line, if the pair of equations is consistent?
1. Explain the Way of Representing the Linear Equation in Two Variables.
Below given are the steps to represent the linear equation in two variables:
The first step is to define the given variables.
Write any two equations which include two variables.
Represent the equation in slope-intercept form i.c. y= mx + c
Examine the solution by analyzing the points where it intersects.
Formulate the meaning of the point in terms of the problem.