Solve the Pair of Linear Equation

Pair of Linear Equations in Two Variables Graphical Method

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 the 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 variables 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:


\[y = \frac{14 - 7x}{3}\]


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:

  1. The graphical method

  2. The algebraic method

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


\[a_{1}x + b_{1}y + c_{1} = 0\] and \[a_{2}x + b_{2}y + c_{2} = 0\]


Here, the numbers \[a_{1}, b_{1}, c_{1}, a_{2}, b_{2} {\text{and}} c_{2}\]are the real numbers.


Also, \[a_{1}^{2} + b_{1}^{2} \neq 0\] and \[a_{2}^{2} + b_{2}^{2} \neq 0\].


If the pair of linear equations is given in these two forms of \[a_{1}x + b_{1}y + c_{1} = 0\] and \[a_{2}x + b_{2}y + c_{2} = 0\], then three different conditions arise:

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

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

  3. 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:

  1. Two lines intersecting with each other

  2. Two lines parallel to each other

  3. 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:

  1. 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.

  1. 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.

  1. Cross-Multiplication Method:

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


\[a_{1}x + b_{1}y + c_{1} = 0\] ...(1)

\[a_{2}x + b_{2}y + c_{2} = 0\] ...(2)


In this method, you multiply the equation (1) by the coefficient of \[b_{2}\] or \[a_{2}\] and in the equation (2) by that of the \[b_{1}\] or \[a_{1}\] 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:


x

2

4

6

8

y = 16 - x

14

12

10

8


x

2

4

6

8

y = x - 4

-2

0

2

4


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

FAQ (Frequently Asked Questions)

1. What do you Mean by a Nonlinear Equation?

A system of the nonlinear equations is the one that deals with two or even more equations in two or more variables which contain a minimum of one equation which is not linear. It creates a straight line or presents the equation for a straight line. It does not form any straight line, however, it forms a curve.

2. How to Solve a Pair of Linear Equations in Two Equations?

The steps to solve a pair of linear equations in two equations are:

  1. First, you need to frame the two linear equation in two different variables

  2. You can solve them either using the elimination, the substitution or the cross multiplication method.

  3. You can also solve them in a graphical manner. You need to draw the lines on a graph by using the technique explained above and then finding the point of intersection

  4. Make sure to always validate the solutions and use both the equations at the end of your answer.