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Does the generic solution to a dependent system always have to be written in terms of x ?

No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of x and if needed x and y .

Solve the following system.

     x + y + z = 7 3 x 2 y z = 4 x + 6 y + 5 z = 24

Infinite number of solutions of the form ( x , 4 x −11 , −5 x + 18 ) .

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Access these online resources for additional instruction and practice with systems of equations in three variables.

Key concepts

  • A solution set is an ordered triple { ( x , y , z ) } that represents the intersection of three planes in space. See
    [link] .
  • A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. See [link] .
  • Systems of three equations in three variables are useful for solving many different types of real-world problems. See [link] .
  • A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. See [link] .
  • Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
  • A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. See [link] .
  • Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.

Section exercises

Verbal

Can a linear system of three equations have exactly two solutions? Explain why or why not

No, there can be only one, zero, or infinitely many solutions.

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If a given ordered triple solves the system of equations, is that solution unique? If so, explain why. If not, give an example where it is not unique.

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If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If not, give an example.

Not necessarily. There could be zero, one, or infinitely many solutions. For example, ( 0 , 0 , 0 ) is not a solution to the system below, but that does not mean that it has no solution.

     2 x + 3 y −6 z = 1 −4 x −6 y + 12 z = −2        x + 2 y + 5 z = 10

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Using the method of addition, is there only one way to solve the system?

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Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.

Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable.

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Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
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What is microbiology
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studies of microbes
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when we takee the specimen which lumbar,spin,
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How bacteria create energy to survive?
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Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
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the significance of food webs for disease transmission
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food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
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Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
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This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
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Examples of thermophilic organisms
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Give Examples of thermophilic organisms
Shu
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Prevent foreign microbes to the host
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they provide healthier benefits to their hosts
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They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
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cell is the smallest unit of life
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cell is the smallest unit of life
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Innocent
cell is the structural and functional unit of life
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is the fundamental units of Life
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There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
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Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
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skin
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skin,Oral,Nasal,GIt
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part of a tissue or an organ being wounded or bruised.
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Binomial nomenclature
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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