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For which values of k does the system of linear equations have zero, one, or an infinite number of solutions? [Note: Not all three possibilities need occur.] 2x1- x2= 3 , 4x1 - 2x2= k

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Answer:

  • for k=6, the system has infinitely many solutions
  • for k≠6, the system has no (zero) solution

Step-by-step explanation:

We have the equations:

  1. [tex]2x_{1} -x_{2} =3[/tex]  
  2. [tex]4x_{1}-2x_{2} =k[/tex]

If we multiply both sides of (1) by 2, we get [tex]4x_{1}-2x_{2} =6[/tex]

We also have from (2) [tex]4x_{1}-2x_{2} =k[/tex]

Thus if k=6, every pair ([tex]x_{1},x_{2}[/tex]) satisfy the equation

if k≠6, the system does not have solution.

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