Pair of Linear Equation in Two Variables

An equation in the form of px + qy + r, where p, q, and r are real numbers and the variables p and q are not equivalent to zero, is known as a linear equation in two variables. 

The two linear equations with similar two variables are considered as the pair of linear equations in two variables. There are two methods to solve a pair of linear equations in two variables. These two methods are:

1. Graphical method

2. Algebraic Method

The general form of pair of linear equation in two variables is given as:

p1x + q1y+ r1 = 0  p2x + q2y + r2 = 0

Here, p1, q1, r1, p2, q2, r2 are real numbers

P12 + q12  ≠ 0

p22 + q22 ≠ 0

Pair of Linear Equation in Two variables Introduction

We have studied linear equations in one variable in our previous classes and we know how to solve the linear equation in one variable. Solving the linear equation in one variable is simple and can be calculated easily. In order to solve the linear equations in two variables, two different sets are needed which will be helpful to determine the values of two unknown variables, say x and y. If one equation is given and two variables are asked to be solved, we will not be able to get a particular solution for a given equation.

For Example

2x + 5y = 6 and 5x + y = 2

We can get a particular solution for the above two equations. But, on the other side

We cannot get a particular solution for the equation 3x + 6y = 5 as only one equation is given and we are asked to find the values of two unknown variables. Hence, we can express the equation 3x + 6y = 5 as, y = (5-3x)/6

The values of y will change accordingly on the basis of the values of variable x, Hence, the particular solution for a given equation is not possible.

It can be said that two distinct sets of independent conditions are required to obtain a particular solution of systems of linear equations in two variables

Expression of Pair of Linear Equation in Two Variables

There are generally two methods to solve and express the pair of linear equations in two variables. These two methods are:

• Graphical Method

• Algebraic Method

The general representation of pair of linear equation for two variables say x and y is given by:

p₁x + q₁y + r₁ = 0………….(1)

p₂x + q₂y + r₂ = 0 …………(2)

Where, p₁ q₁, r₁, p₂, q₂, r₂ are real numbers and p₁² + q₁² ≠ 0  and p₂²+ q₂² ≠ 0.

If the pair of linear equations are given in the form of p₁x + q₁y + r₁ = 0 and p₂x + q₂y + r₂ = 0, then the following situation takes place.

1. If the given pair of linear equations is consistent then p₁/p₂ ≠ q₁/q₂

2. If the given pair of linear equations is inconsistent then p₁/p₂ = q₁/q₂ ≠ r₁/r₂

3. If the given pair of linear equations is dependent and consistent then p₁/p₂ = q₁/q₂ = r₁/r₂.

For example, the two equations such as 2x- y = -1 and 3x + 2y = 9 are pairs of linear equations with two variables x and y. The diagram given below represents the solutions for both the equations by substituting the value of x to determine the value of y.

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(Image will be uploaded soon)

Similarly, we can find the solutions for other pairs of linear equations in two variables.

Graphical Expression of Pair of Linear Equation in Two Variables 

As we know, the graphical representation of linear equations in two variables always forms a straight line. Therefore, a pair of linear equations in two variables forms a straight line which is observed together. There are three possibilities when two lines are drawn in a plane. These three possibilities are :

• Two lines intersect with each other at one point.

• Two lines which are drawn are parallel to each other.

• The two lines will coincide with each other.

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Now, let us take a pair of linear equations such as x + y + 5 and 2x + 2y = 10. To represent these pairs of linear quotations in a graph, we need to determine the solutions.

We will get two different values of x and y after determining the solutions. Hence it can be expressed as below:

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(Image will be uploaded soon)

Algebraic Method of Solving a Pair of Linear Equations of Two Variables.

Pairs of linear equations are found in every situation. Let us learn the algebraic method of solving a pair of linear equations of two variables by considering the practical application of daily life. For example, suppose you went to the fruits market. There were two types of oranges available. The fruit seller said that the small oranges are 3 times smaller than the large oranges. The total money the lady was having with her is Rs.100. 

Can you determine how much the lady spent to purchase two types of oranges?

Let  us understand the above situation mathematically,

Let us consider the price of the smaller oranges = x

And, the price of the bigger oranges = y.

As per the first situation in the questions   x = 4y…………(1)

And as per the second situation in the questions, x + y = 100……………(2)

For determining the solution, we are required to find the value of both x and y and solve both equations. The coordinates (x,y) can be represented easily in a graph. But the graphical method is the easiest way in the situations when the point representing the solution of the linear equations includes non-integral coordinates like (√2, 5√6), (–1.42, 5.3), (6/19, 1/17), etc. Hence, algebraic methods are used to solve such situations.

The two algebraic methods to solve a pair of linear equations:

1. Substitution Method

2. Elimination Method

3. Cross-Multiplication Method

Solved Examples

1. The sum of 4 times a larger integer and 5 times a smaller integer is 7. When twice the smaller integer is subtracted from 3 times the larger, the result is 11. Find the integers.

Solution:

We will first assign the variables to the larger and smaller integer.

Let us consider x as the smaller integer and y as the larger integer.

When Using two variables, we are required to form two equations. The first sentence explains the addition and the second statement explains the difference.

This gives the following situation

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(Image will be uploaded soon)

This gives us the following two equations.

4x + 5y = 7

3x- 2y = 11

We will solve the above equation through the substitution method.

In the substitution method, we will multiply the equation 4x + 5y = 7  by -2 and  the equation 3x - 2y = 11 by 5

Hence, two equations that we got after multiplication  are given below:

-8x -10y = -14……(1)

15x - 10y = 55…..(2)

Now, we will subtract the equation (2) from the equation (1)

(15x - 10y =  55) - (-8x -10y = -14)

23x = 69

x = 69/23

x = 3

Now, we will substitute the value of x to find the value of y

4x + 5y = 7

4(3) + 5y = 7

12 + 5y = 7

5y = -5

y = -1

Hence, the larger integer is 3 and the smaller integer is -1

2. Calculate  the value of variables that meets the following equation

2x + 5y = 20 and 3x+6y =12.

Solution:

We will solve the pair of linear equations by the substitution method.

Two equations are given below:

2x + 5y = 20…………………….(1)

3x + 6y =12……………………..(ii)

Now, we will multiply the equation (i) by 3 and equation (ii) by 2, 

After multiplication, we get the following two equations

6x + 15y = 60…………………….(iii)

6x+12y = 24……………………..(iv)

Now , we will Subtract the equation (iv) from (iii)

(6x+12y = 24) - (6x + 15y = 60) = 0

-3y = - 36

y = 12

By substituting the value of y in any of the equation (i) or (ii), we get the value of x

2x + 5(12) = 20

x = −20

Hence x=-20 and y =12 are two-point where the given equations intersect

What is a Linear equation?

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 = (9-6x)/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:

1. Graphical Approach

2. 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 and a12 + b12 0

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