Solve the Pair of Linear Equations: Methods and Solutions

A linear equation in two variables is an equation in the form ax+by+c, where a,b, and c are real values and a,b are not equal to zero. We deal with two such equations in a pair of linear equations in two variables. A point on the line representing the equation is the solution of such equations.

An equation is a two-sided expression that has the same sign on both sides. A polynomial is a mathematical expression having non-negative integer powers for the variables. For instance, x⁴ + 3x³ + 2x⁹ is a polynomial, whereas x³/5+ 3x0.6 is not. We should be aware of the concept of 'degree' when defining polynomials. The highest power of the variable in the given polynomial is defined as a degree. A linear polynomial is a polynomial with degree one. A polynomial of degree 2 is referred to as a quadratic polynomial, whereas a polynomial of degree 3 is referred to as a cubic polynomial.

How to solve the Pair of Linear equations?

Here, we cannot get a particular solution for this as there is only one condition given, and we have two unknowns. We can rewrite the above equation as:

y = \[\frac {(9-6 x)} {7}\]

The values of y will change in accordance with the values of x. As a result, there is no such thing as a one-of-a-kind answer.

As a result, it is evident that in order to obtain a specific solution of a system of linear equations in two variables, two alternative sets of independent conditions are required.

Representation

Two methods can be used to solve and express the pair of linear equations:

Graphical Approach

The Algebraic Approach

The general formulation of a set of linear equations in two variables, say x and y, is:

 a₁x + b₁y + c₁ = 0..............(1)

 a₂x + b₂y + c₂ = 0..............(2)

where a₁, b₁, c₁, a₂, b₂, c₂, a¹₂ + b¹₂ = 0 and a²₂+ b²₂ = 0 are all real values 

When the pair of linear equations is written as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, three conditions apply:

1. An inconsistent pair of linear equations is a set of two linear equations in two variables that has no solution.

2. A consistent pair of linear equations in two variables is a pair of linear equations with a solution.

3. A dependent pair of linear equations is a pair of linear equations in two variables that has an infinite solution.

Graphical Representation

If there are two lines in a plane, there are three possibilities:

1. The two lines cross each other,

2. Are parallel to one other

3. Or coincide with each other.

Algebraic Representation

Let's begin by looking at how we might use simultaneous linear equations in our daily lives. In any potential case, a pair of linear equations can be found. Let's pretend you went to the fish market to buy some fish. The fish were available in two sizes. According to the fisherman, the total price of the smaller fish is three times that of the larger fish. In addition, the total amount of money you purchased from your residence is Rs.100. Can you tell me how much you paid for the two different types of fish?

Let's look at this from a mathematical standpoint.

 Allow for a pricing difference of Rs.x for the smaller fish and Rs.y for the larger fish.

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

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

To discover the solution, we must solve both equations and determine the values of x and y. In a graph, the coordinates (x, y) can be simply arranged. However, when the point representing the solution of linear equations has non-integral coordinates, such as (3, 2 7), (–1.75, 3.3), (4/13, 1/18), etc., the graphical method is inconvenient. As a result, we tackle such problems using algebraic approaches.

To solve a pair of linear equations, you can use the algebraic methods listed below:

1. Substitution Techniques

2. Method of Elimination

3. The Method of Cross-Multiplication

A linear equation in two variables refers to the equation in the form of ax + by + c, wherein a,b and c are the real numbers and a,b and c are not equal to zero. However, in the pair of linear equations with two variables, you have to deal with two of these equations.  The solution of these equations is a point on the line which represents the equation. An equation is called to be an expression when it has an equality sign on both sides. A polynomial consists of a mathematical expression having powers of the variables as the non-negative integers. When you define polynomials you must know about the concept of the degree. A degree is defined as the highest power of a variable in a given polynomial. A polynomial having a degree 1 is said to be a linear polynomial. A polynomial having a degree 2 is said to be a quadratic polynomial and similarly, a polynomial having a degree 3 is called a cubic polynomial. In this article, we will learn how to solve the pair of linear equations and the pair of linear equations in two variable graphical methods.

Solving the Pair of Linear Equation in Two Variables

You have studied the linear equation in one variable and you would know how to solve it. If there was just one variable and just one equation, you could solve it easily, but in this case, you have two different variables and two different equations. Here you need two different sets of the linear equations for finding out the two different unknowns. If one equation is given to you and you have to solve two variables you will not find a particular solution.

Consider for example the given equations:

4x + 2y = 10 and 5x + 2y = 15

You can solve these simultaneous equations and you can arrive at a particular solution from these equations but on the other hand, if you consider,

7x + 3y = 14

In this case, you cannot get a particular solution as there is only one condition given and you have two different unknowns. You can rewrite the above equation in the following way:

y=14−7x³

Depending on the different values of x, the values of y will also change accordingly. Hence, one unique solution is not possible in this case.

Thus, you can clearly say that to get a specific solution of the systems of linear equations in two variables, you need to have two different sets of independent conditions.

Representation of the Pair of Linear Equation in Two Variables

You can solve the pair of linear equations and represent it in two different ways:

• The graphical method

• The algebraic method

The general representation of the pair of linear equation in two variables x and y is denoted as:

a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0

Here, the numbers a₁,b₁,c₁,a₂,b₂ and c₂ are the real numbers.

Also, a²₁+b²₁≠0 and a²₂+b²₂≠0

.If the pair of linear equations is given in these two forms of a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0, then three different conditions arise:

The pair of linear equations in two variables that have no solution is referred to as an inconsistent pair of the linear equations.

The pair of linear equations in two variables that has a solution is called a consistent pair of the linear equations.

The pair of linear equations in two variables that has an infinite solution is said to be a dependent pair of the linear equations.

Pair of Linear Equations in Two Variables Graphical Method

If two lines are present in a plane, there could be three different possibilities that are as follows:

Two lines intersecting with each other

Two lines parallel to each other

Two lines coinciding each other

You can represent these conditions graphically as follows:

(images will be uploaded soon)

Pair of Linear Equations in Two Variables Algebraic Method

You can solve a pair of linear equations in two variables through the algebraic method in the following ways:

Substitution Method:

You substitute one of the given equations in the other by substituting one of the variables in the form of another. Now, the equation would consist of only one variable and you can then solve it accordingly to get the result.

Elimination Method: 

As the name suggests, the elimination method refers to the elimination of one of the variables from the given set of the equations. Solving it would give you the desired result.

Cross-Multiplication Method:

The general form of the pair of linear equations in two variables is given as follows:

a₁x+b₁y+c₁=0 ...(1)

a₂x+b₂y+c₂=0 ...(2)

In this method, you multiply the equation (1) by the coefficient of b₂ or a₂ and in the equation (2) by that of the b₁ or a₁ and then eliminate one of the variables and solve. The name is given because the multiplication in the equations having the coefficients of the variables are carried in a cross-multiplication fashion.

Solved Examples

Let us now look at some of the solved examples:

Example 1: Find the values of the two variables that satisfy the following equations:

2x + 5y = 20

3x + 6y = 12

Solution: Using the substitution method to solve the pair of linear equations you have:

2x + 5y = 20 . . . (i)

3x + 6y = 12 . . . (ii)

Multiplying the equation (i) by 3 and the equation (ii) by 2, you get

6x + 15y = 60 . . . (iii)

6x + 12y = 24 . . . (iv)

Subtracting the equation (iv) from (iii),

3y = 36

Hence, y = 12

Substituting the value of the variable y in any of the above-given equation (i) or (ii), you get,

2x + 5(12) = 20

Hence, x = -20

Hence, x = -20 and y = 12 is the point in a plane where the given equations would intersect.

Example 2: Solve the given two equations graphically:

x + y = 16

x - y = 4

Solution: The solutions for each of the equations is as follows:

From the graph, you can find the common point of intersection which is (10, 6).