## SOLVING SYSTEM OF LINEAR EQUATIONS BY RANK METHOD

Step 1 :

Find the augmented matrix [A, B] of the system of equations.

Step 2 :

Find the rank of A and rank of [A, B] by applying only elementary row operations.

Note :

Column operations should not be applied.

Step 3 :

Case 1 :

If there are n unknowns in the system of equations and

ρ(A)  =  ρ([A|B])  =  n

then the system AX = B, is consistent and has a unique solution.

Case 2 :

If there are n unknowns in the system AX = B

ρ(A)  =  ρ([A| B]) < n

then the system is consistent and has infinitely many solutions and these solutions.

Case 3 :

If ρ(A)  ≠  ρ([A| B])

then the system AX = B is inconsistent and has no solution.

Example 1 :

Solve the following linear equation by rank method

2x + 5y + 7z  =  52, x + y + z  =  9, 2x + y - z  =  0

Solution :   Number of non zero rows are 3.

ρ(A)  =  ρ([A|B])  =  3. The system is consistent and it has unique solution.

From 1st row,

x+y+z  =  9 -----(1)

From 2nd row,

3y+5z  =  34 -----(2)

From 3rd row,

-4z  =  -20 -----(3)

From (3)

z  =  5

By applying the value of z in (2), we get

3y+5(5)  =  34

3y + 25  =  34

3y  =  34-25

3y  =  9

y  =  3

By applying the value of y and z in (1), we get

x+3+5  =  9

x+8  =  9

x  =  1

So, the solution is (1, 3, 5).

Example 2 :

Solve the following linear equation by rank method

4x - 2y + 5z  =  6, 3x + 3y + 8z  =  4,  x - 5y - 3z  =  5

Solution :   ρ(A)  =   2 and ρ([A|B])  =  3. The system is inconsistent and it has no solution.

Example 3 :

Solve the following linear equation by rank method

x+9y-z  =  27, x-8y+16z  =  10, 2x+y+15z  =  37

Solution :  Here ρ(A)  =  ρ([A|B])  =  2 < 3, then the system is consistent and it has infinitely many solution.

From the 1st row,

x + 9y-z  =  27    ---(1)

From the 2nd row,

17y + 17z  =  -17   ---(2)

Dividing by 17, we get

y + z  =  -1

Put z  =  t

y  =  -1 - t

By applying the value of y and z in (1), we get

x + 9(-1-t)-t  =  27

x - 9+9t-t  =  27

x  =  27+9-8t

x  =  36-8t

Solution :

x  =  36-8t, y  =  -1-t and z  =  t where t ∈ Real numbers. Apart from the stuff given aboveif you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

1. Click on the HTML link code below.

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test 