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System of Equations

Set of two or more equations which contain same variables is known as a system of equations.


The equations a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0 make a system of equations.

Methods of solving a system of simultaneous linear equations

  • Method of substitution
    Step-1. Find the value of y in terms of the other variable x from any one of the two equations.
    Step-2. Substitute this value of y obtained in Step-1 in the other equation. Then, solve this equation to get the value of x.
    Step-3. Substitute the value of x obtained in Step-2 in any of the given equations and find the value of y.
    Alternately, we may find x in terms of y. Then, we shall get the value of y from Step-2 above and value of x from Step-3 above.

    Solve for x and y:
    Description: 38331.png 

    Let’s first find x in terms of y from equation (1) Description: 38319.png 
    Substitute the value of x in (2)
    Description: 38313.png 
    Substitute the value of y in (1)
    Description: 38307.png 
  • Method of Elimination
    As the name suggests, this method involves eliminating one of the variables from the given system of equations. Then, we solve to get the value of the other variable and substitute in any one equation to get the value of eliminated variable.
    Step-1. Find the L C M of coefficients of one variable (say x) in the given equations. Multiply the equations by a suitable number such that coefficients of x are numerically equal (to the L C M ) in both the equations.
    (a) If the coefficients of variable (x) have the same sign, then subtract the equations obtained in Step-1.
    (b) If the coefficients have opposite sign, then add the two equations.
    Step-3. Solve the equation obtained in Step-2 as the linear equation of one variable y.
    Step-4. Substitute the value of y obtained in the above step in any of the equations to obtain the value of the other variable (x).
    The same procedure can be adopted if we decide to eliminate y instead of x.
    Solve Description: 38301.png
    Let Description: 38295.png……………(1)
    and Description: 54713.png…………… (2)
    The coefficients of x in the above equations are 8 and 3. The L C M of 8 and 3 is 24. Therefore, Eq. (1) has to be multiplied by 3 and Eq. (2) has to be multiplied by 8 to make the coefficients of x equal.
    Description: 38289.png…………… (3)
    Description: 54602.png…………… (4)
    Subtracting, Eq. (3) – Eq. (4)
    Description: 38283.png
    Substitute for y in Eq. (1), we get
    Description: 38277.png
  • Method of cross-multiplication
    This method can be used for a system of equations which is consistent. For dependent and inconsistent systems, we cannot use it.
    Follow the steps given below:
    Step-1. Write down the given system of equations as shown below. Bring all the terms on one side leaving the right hand side to be zero.
    a1 x + b1 y + c1 = 0
    a2 x + b2 y + c2 = 0
    Step-2. Write the coefficients, starting with the coefficients of y, then the constant terms, followed by the coefficients of x, and repeat the coefficients of y as shown below:
    Description: 38271.png
    Step-3. Write the coefficients of x, y and 1 in the following manner:
    Description: 38263.png 
    Description: 38257.png
    Below x, we write the coefficients of y and the constant terms. Below y, we write the coefficients of x and constant terms. Below 1, we write the coefficients of x and y.
    From Step-3 we get.
    Description: 38251.png 
    We find that
    (a) Description: 38245.png
    (b) Description: 38239.png 
    So, the system has no solution as division by zero is not defined.
    (c) Description: 38233.png
    Such solutions are dependent and the given equations have infinite number of solutions.

    Solve Description: 38227.png
    Description: 38221.png
    Eq. (1) and Eq. (2) can be written as
    Description: 38215.png 
    Description: 38208.png 
    Description: 38202.png 
    Description: 38196.png 

Method of solving simultaneous linear equation with 3 variables

There are two methods of finding the solution of a system of linear equation in three variables they are:

  1. Method of elimination
  2. Method of cross-multiplication

    The methods are similar to the one already discussed for the solution system of linear equation in two variables.

Solve for xy and z:
Description: 38189.png 
(a) Method of elimination
2x – y + z = 3………………….(1)
x + 3y – 2z = 11…………………. (2)
3x – 2y + 4z = 1…………………. (3)
By Eq. (1) × 2 we get
4x – 2y + 2z = 6…………………. (4)
By Eq. (2) + Eq. (4), 5x + y = 17……(5) [the variable z is thus eliminated]
By Eq. (2) × 2,        2x + 6y – 4z = 22…………………. (6)
By Eq. (3) + Eq. (6),     5x + 4y = 23…………………. (7)
By Eq. (5) – Eq. (7),      –3y = – 6 or y = 2
Putting y = 2 in Eq. (5),     5x + 2 = 17, or 5x = 15 or, x = 3
Putting x = 3 and y = 2 in Eq. (1)
2 × 3 – 2 + z = 3
or 6 – 2 + z = 3
or 4 + z = 3
or z = –1
So, x = 3, y = 2, z = –1 is the required solution.
(Any two of 3 equations can be chosen for elimination of one of the variables.)
(b) Method of cross-multiplication
We write the equations as follows:
2x – y + ( z – 3) = 0
x + 3y + (–2 z –11) = 0
By cross-multiplication,
Description: 38183.png 
Description: 38177.png 
Description: 38171.png 
Substituting above values for x and y in Eq. (3), i.e., 3x – 2y + 4z = 1, we have
Description: 38165.png 
or 60 – 3z – 10z – 38 + 28z = 7
or 15z = 7 – 22 or 15z = –15 or z = –1
Now Description: 38159.png 
Thus, x = 3, y = 2, z = –1


Two numbers are such that sum of five times the first and thrice the second is 7. Also, the difference of three times the first and twice the second is 8. Find the numbers.
Both the numbers are unknown. Let’s assume the first number as x and the second numbers as y.
As given in the question, the first condition says,
5x + 3y = 7…………………(1)
The second condition says,
3x – 2y = 8………………… (2)
Now, we have two linear equations in two variables. We can apply any of the 3 methods to solve these equations. Let us use elimination method here.
Coefficients of x in the equations are 5 and 3. L C M of 5 and 3 is 15.
We can multiply the first equation by 3 and second equation by 5 to make both the coefficients equal to 15.
Description: 38153.png………………… (3)
Description: 38147.png………………… (4)
Now, coefficients of x in Eqs. (3) and (4) are equal. So, we can subtract Eq. 4 from Eq. 3 to eliminate x and find y.
Description: 38141.png 
Description: 38135.png 
Description: 38129.png 
We can find x by substituting this value of y in any of the given equations. Substituting in Eq. (1) we get,
Description: 38123.png 
Description: 38117.png 


In a fraction, if 1 is added to the denominator, the fraction becomes 1/2. If 2 is added to the numerator, the fraction becomes 1. Find the fraction.
Both numerator and denominator of the fraction are unknown. Let’s assume numerator to be x and denominator to be y. The fraction becomes Description: 38110.png.
The first condition can be written as Description: 38104.png.
This gives us Description: 38098.png 
The second condition can be written as Description: 38092.png 
Description: 38086.png 
Now, we have a system of linear equations which can be solved using any of the three methods.
Coefficient of y in both equations is –1. We can directly subtract Eq. (2) from Eq. (1) to eliminate y.
Description: 38080.png 
Substituting this value of x in Eq. (2), we get,
Description: 38074.png 
Hence, the required fraction becomes Description: 38068.png.

Consistent and inconsistent systems of two simultaneous linear equations

A system having two simultaneous linear equations is said to be consistent if it has at least one solution (i.e., it has one or more than one solution).

A system having two simultaneous linear equations is said to be inconsistent if it has no solution.

Consider two simultaneous linear equations, say

Description: 38062.png 

Description: 38056.png 


Hence, it is consistent and the graph is a pair of intersecting lines.


Description: 38050.png 


Hence, it is consistent and the graph is a pair of overlapping lines.


Description: 38044.png 


Hence, it is inconsistent and the graph is a pair of parallel lines.

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