# SOLVING LINEAR EQUATIONS WITH 3 VARIABLES USING RANK METHOD

Solving Linear Equations in Three Variables Using Rank Method :

• 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.
• If there are n unknowns in the system AX = B, and ρ (A) = ρ ([A| B]) = n − k, k ≠  0 then the system is consistent and has infinitely many solutions and these solutions form a k − parameter family. In particular, if there are 3 unknowns in a system of equations and ρ (A) = ρ ([A| B]) = 2, then the system has infinitely many solutions and these solutions form a one parameter family.
• In the same manner, if there are 3 unknowns in a system of equations and ρ (A) = ρ ([A| B]) = 1, then the system has infinitely many solutions and these solutions form a two parameter family.
• If ρ(A)  ρ ([A| B]), then the system AX = B is inconsistent and has no solution.

Problem 1 :

Solve the following linear equations by rank method

2x + y + z  =  5, x + y + z  =  4, x - y + 2z  =  1

Solution :

 2 1 1 5 1 1 1 4 1 -1 2 1

ranking method examples 1

˜

 2 1 1 5 1 1 1 4 1 -1 2 1

R₂ <-> R₂

˜

 1 1 1 4 2 1 1 5 1 -1 2 1

R₂ => R₂ - 2R₁

2          1         1         5

-2         -2       -2       -8

----------------------------

0        -1       -1       -3

R₃ => R₃ - R₁

1         -1         2         1

1          1         1         4

---------------------------

0        -2        1        -3

ranking method examples 1

˜

 1 1 1 4 0 -1 -1 -3 0 -2 1 -3

R₃ => R₃ - 2R

0        -2         1         -3

0        +2        +2      +6

----------------------------

0         0          3         3

˜

 1 1 1 4 0 -1 -1 -3 0 0 3 -3

R₃ => R₃ - 2R₂

Rank (A)  =  3

Rank [A,B]  =  3

If rank (A)  =  rank of [A,B]  =  number of unknowns then we can say that the system is consistent and it has unique solution.

x + y + z  =  4   --------(1)

-y - z  =  -3  --------(2)

3z  =  3 --------(3)

z  =  1

Apply z = 1 the second equation to get the value of y

- y - 1  =  -3 and y  =  2

Apply z  =  1 and y  =  2 in the first equation to get the value of x

x + 3  =  4

x  =  1

Solution :

x = 1,  y = 2 and z = 1

Apart from the stuff given above, if 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

We always appreciate your feedback.

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

Word problems on quadratic 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