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System of Linear Equations

System of Linear Equations

Edited By Komal Miglani | Updated on Oct 11, 2024 12:28 PM IST

In mathematics, a system of linear equations, also known as a linear system, consists of one or more linear equations that involve the same set of variables. There are many methods by which we can solve the system of linear equations. In real life, we use the system of linear equations to solve age-related problems and time-related problems.

This Story also Contains
  1. System of Linear EquationA system of linear equations are group of n linear equations containing n number of variables.
  2. 1. System of 2 Linear Equations:
  3. 2. System of 3 Linear Equations:
  4. Consistent system of equations:
  5. Inconsistent equation:
  6. Homogenous System of Linear Equation
  7. Non Homogenous System of Linear Equations
  8. Methods to solve Systems of Linear Equations in two variables
  9. Methods to solve Systems of Linear Equations in three variables
  10. Solved Examples Based on System of Linear Equations
System of Linear Equations
System of Linear Equations

In this article, we will cover the concept System of linear equations. This category falls under the broader category of Matrices, which is a crucial Chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Questions based on this topic have been asked frequently in JEE Mains

Background wave

System of Linear Equation
A system of linear equations are group of n linear equations containing n number of variables.

1. System of 2 Linear Equations:

It is a pair of linear equations in two variables. It is usually of the form

a1x+b1y+c1=0

a2x+b2y+c2=0

Finding a solution for this system means finding the values of x and y that satisfy both equations.

2. System of 3 Linear Equations:

It is a group of 3 linear equations in three variables. It is usually of the form

a1x+b1y++c1z+d1=0

a2x+b2y++c2z+d2=0

a3x+b3y++c3z+d3=0

Finding a solution for this system means finding the values of x,y, and z that satisfy all three equations.


The system of equations is broadly classified into two types:

Consistent system of equations:

A system of equations is said to be consistent if it has at least one solution. Let the given system of equations is

The system of linear equationsa1x+b1y=c1a2x+b2y=c2has exactly one solution ifa1a2b1b2.

 E.g., x+y=2xy=6 is consistent because it has a solution x=4 and y=2.
Given lines are non-parallel, hence lines will have one point of intersection.

It has infinite solutions ifa1a2=b1b2=c1c2.

In this case, two lines represented by these lines coincide, so there are infinite pair of values of x and y that satisfy both the equations. This case is also counted as consistent as there is at least one solution.

Inconsistent equation:

A system of equations is said to be inconsistent if it has no solution.

Leta1x+b1y+c1=0anda2x+b2y+c2=0,thenequationareinconsistentandhasnosolutionifa1a2=b1b2c1c2


For example, x+y=5 and 2x+2y=5 are inconsistent as it has no solution, just by seeing the equation, we get that it is the equation of two different parallel lines which never intersect. These lines are non-intersecting, hence there is no solution to this system.

Homogenous System of Linear Equation

A linear equation with a constant value of zero is called a homogeneous equation.

Let,a1x+b1y+c1z=0...(i)a2x+b2y+c2z=0...(ii)a3x+b3y+c3z=0...(iii)bethreehomogeneousequationsandletΔ=|a1b1c1a2b2c2a3b3c3|

Note that x=y=z=0 will always satisfy this system of equations. So system of homogeneous equations will always have at least one solution.

Also, the solution x=0,y=0, and z=0 is called a trivial solution, and other solutions are called non-trivial solutions.

  • If Δ0, then x=0,y=0,z=0 is the only solution of the above system. This solution is also known as a trivial solution.
  • If Δ=0, at least one of x,y, and z are non-zero. In this case, we will have non-trivial solutions as well. Also, there would be infinite solutions of such a system of equations.

Non Homogenous System of Linear Equations

A linear equation with a constant value not equal to zero is called a homogeneous equation.

Characteristics of non Homogenous Linear Equations

  • If |A|0, then the system of equations is consistent and has a unique solution X=A1 B
  • If |A|=0 and (adjA)B0, then the system of equations is inconsistent and has no solution.
  • If |A|=0 and (adjA)B=0, then the system of equations is consistent and has an infinite number of solutions.
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Methods to solve Systems of Linear Equations in two variables

We use the following method to solve a System of linear equations in two variables

  • Graphical method
  • Elimination method
  • Substitution method
  • Matrix method
  • Cross Multiplication method

Methods to solve Systems of Linear Equations in three variables

We use the following method to solve a System of linear equations in three variables

  • Cramers Rule
  • Inverse method
  • Gaussian elimination method
  • Gaussian Jordan method
  • LU Decomposition method

Summary

The system of linear equations plays a vital role in solving many varieties of problems. We can use a System of linear equations to solve Speed, distance, and time problem as well as wages problem. Knowing the type of equation helps us to solve the equation in simpler ways. It also us in finding the properties of equations.

Recommended Video :

Solved Examples Based on System of Linear Equations

Example 1:

Let the system of linear equations

x+2y9z=7x+3y+7z=92x+y+5z=83x+y+13z=7
has a unique solution x=α,y=β,z=γ Then the distance of the point (α,β,γ) from the plane 2x2y+z=λ. is [JEE MAINS 2023]

Solution

x+2y9z=7(1)x+3y7z=9(2)2x+y+5z=8(3)(2)(1)y+16z=2(4)(3)2x(1)3y+23z=6(5)3x(4)+(5)71z=0z=0y=2x=3(3,2,0)(a,β,γ) Put in 3x+y+13z=1λ=9+2=11d=|6411|=?3

Hence, the answer is 3.

Example 2: Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations

x+y+z=12x+Ny+2z=23x+3y+Nz=3

has a unique solution is k6 then the sum of the value of k and all possible values of N is [JEE MAINS 2023]

Solution

For unique solution

Δ0|1112 N233 N|0(N26)(2 N6)+(63 N)0N25 N+60

N3& N2

 Hence N can be {1,4,5,6} Fav case : 46=K6k=4

sum=20

Hence, the answer is 20.

Example 3:

If the system of equations

2x+yz=52x5y+λz=μx+2y5z=7

has infinitely many solutions, then (λ+μ)2+(λμ)2 is equal to: [JEE MAINS 2023]

Solution

Δ=0|21125λ125|=02(252λ)1(10λ)1(4+5)=0513x=0λ=17Δx=0|511μ517725|=0

5(2534)1(5μ119)1(2μ+35)=045+5μ+1192μ35=039+3μ=0μ=13(λ+μ)2+(λμ)2=42+(30)2=916

Hence, the answer is 916.

Example 4 :

Let S denote the set of all real values of λ such that the system of equations

λx+y+z=1x+λy+z=1x+y+λz=1
λx+y+z=1x+λy+z=1x+y+λz=1
is inconsistent, then λS(|λ|2+|λ|) is equal to

[JEE MAINS 2023]

Solution

Given system of equations is inconsistent

Δ=0|λ111λ111λ|=0λ33λ+2=0(λ1)2(λ+2)=0λ=1,2
But for λ=1 all planes are the same
Then λ=2

λs(|λ|2+|λ|)=4+2=6

Hence, the answer is 6.

Example 5: If the system of linear equations

7x+11y+αz=13

5x+4y+7z=β

175x+194y+57z=361

has infinitely many solutions, then α+B+2 is equal to :

[JEE MAINS 2023]

Solution

7x+11y+αz=13

5x+4y+7z=β
175x+194y+57z=361
Condition of Infinite Many solutions

Δ=0 \& Δx,Δy,Δz=0
check.
After solving we get α+13+2=4

Hence, the answer is 4.


Frequently Asked Questions (FAQs)

1. What is a System of linear equations?

A system of linear equations are group of n linear equations containing n number of variables.

2. What are non Homogeneous Equations?

A linear equation with a constant value not equal to zero is called a homogeneous equation.

3. What is inconsistent equation?

A system of equations is said to be inconsistent if it has no solution.

$\\mathrm{Let \; a_1x +b_1y + c_1 = 0\; and \; a_2x + b_2y + c_2 = 0, then} \\mathrm{equation\; are \; inconsistent \; and\; has \; no\; solution\; if} \\mathrm{\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}}$

4. If Δ=0 , and any of Δ10orΔ20orΔ30 ,then what are the characteristics of the equation?

If Δ=0, and any of

Δ10orΔ20orΔ30 

Then the system of equations is inconsistent and hence no solution exists.

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