Pair of linear equations in two variables

In mathematics, a linear equation in two variables is defined as an equation in the form ax+by+c in which the variables are real numbers and the variables are not equal to zero. A linear equation in two variables has the form ax+by+c when the variables are real numbers. In contrast, when dealing with a pair of linear equations in two variables, we are dealing with two of the same equations. The solution of such equations is represented by a point on the line that represents the solution. 

Notes of Pair of Lnear Equations in Two Variables

We have learned about linear equations in one variable in previous classes, and we are familiar with how to solve them. We used to be able to solve it quickly if there was only one variable and one equation, but in this case there are two variables and two equations to contend with. One thing is certain: in order to find out the two different unknowns, we will need two different sets of linear equations, one for each unknown. If we are given a single equation and are asked to solve it for two variables, we will not get the specific solution we are looking for.

As an illustration,

3x + 2y = 9 and 5x + y = 10 are the answers.

These simultaneous equations can be solved, and we can derive a specific solution from them, but on the other hand, they are difficult to solve.

6x + 7y = 9.

We will not be able to find a specific solution to this problem because there is only one condition given and two unknowns in this situation. We can rewrite the equation above as follows:

y = (9-6x)/7

The values of y will change in response to changes in the values of x and vice versa. As a result, there is no such thing as a singular solution.

Consequently, it is clear that in order to obtain a specific solution of systems, or linear equations in two variables, we must first establish two different sets of independent conditions.

Representation of  pair of linear equations in two variables

It is possible to solve and represent the pair of linear equations using two different methods:

1)Method Utilizing Graphs

2)The Algebraic Approach

It is possible to represent a pair of linear equations in two variables, say x and y, using the following general representation:

a1x + b1y + c1 = 0…………… (1)

a2x + b2y + c2 = 0………….. (2)

in which a1, b1, c1, a2, b2, and c2 are all real numbers, and a12+ b12≠ 0, a22+ b22≠0.

Suppose a pair of linear equations is given in the form of a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, and the first of these conditions is met:

1)If the pair of linear equations is consistent, then the following is true: a1/a2≠b1/b2

2)The following is true if the pair of linear equations is incompatible: a1/a2 = b1/b2≠c1/c2

3)The following equation holds if the pair of linear equations is dependent and consistent: a1/a2 = b1/b2 = c1/c2

For example, the linear equations 2x-y = -1 and 3x + 2y = 9 have the variables x and y as their variables. As you can see in the diagram below, we were able to find solutions for both equations by substituting the value of x into the equation to obtain the value of y.

Similar to this, we can come up with solutions for the following situations:

2x+3y-4 = 0, as well as 3x+2y+4 = 0.

6x=y and 4x-3y-5 = 0 are both true.

But what if we have to graph these equations in order to represent them? Let’s take a look at the next section.

Representation in Graphical Form

You learned in Class 9 that if there are two lines present in a plane, then there are three possible outcomes, such as the following:

It is possible that the two lines intersect, that they are parallel to each other, or that they coincide with each other.

The three conditions listed above can be represented graphically as shown in the diagram below:

Consider the following pair of linear equations: x + y = 5, 2x + 2y = 10, and so on. In order to plot these two linear equations as a straight line on a graph, we must first find the solutions to the linear equations. Following the discovery of the solutions, we obtain the distinct values of x and y. As a result, it can be represented as follows:

Approaches to Solving a Pair of Linear Equations Using Algebraic Methods

Let us begin by discussing the practical application of simultaneous linear equations in our daily lives, which we can see in the figure below. A pair of linear equations has the ability to find its way through any situation. Consider the following scenario: you went to the fish market to purchase fish. The fishes were available in two different sizes. It was revealed by the fisherman that the total price of the smaller fish is three times greater than its counterpart, the larger fish. Additionally, the total amount of money that you purchased from your home is Rs.100. Is it possible for you to find out how much money you spent on the two types of fish?

Let us try to understand this in terms of mathematics.

The price of the smaller fish is set at Rs.x, while the price of the larger fish is set at Rs.y.

According to the first condition, x =3y………. (1)

Furthermore, according to the second condition, x + y = 100.…(2)

In order to find the solution, we must first solve both equations and determine the values of x and y on the same line. The coordinates (x, y) can be easily placed in a graph with no difficulty. While this method is convenient in many situations, it becomes less so in situations where the point representing the solution of the linear equations has non-integral coordinates such as (✓3, 2✓7), (–1.75, 3.3), (4/13, 1/18), and so on. As a result, algebraic methods are employed to resolve such situations.

There are several types of algebraic methods that can be used to solve a pair of linear equations, including:

•Method of Substitution Method of •Elimination

•Method of Multiplication by Crossing

Conclusion

In mathematics, a linear equation in two variables is defined as an equation in the form ax+by+c in which the variables are real numbers and the variables are not equal to zero. A linear equation in two variables has the form ax+by+c when the variables are real numbers. In contrast, when dealing with a pair of linear equations in two variables, we are dealing with two of the same equations. The solution of such equations is represented by a point on the line that represents the solution. It is possible to solve and represent the pair of linear equations using two different methods:(1)Method Utilizing Graphs(2)The Algebraic Approach

It is possible to represent a pair of linear equations in two variables, say x and y.There are several types of algebraic methods that can be used to solve a pair of linear equations, including:

•Method of Substitution Method of •Elimination

•Method of Multiplication by Crossing.