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I need a little help! Can someone check my work? It's a written response with linear systems.

Task 1: You are given a linear system of two equations in two unknowns. Before solving, describe how can you mentally check whether the system is independent and consistent? In which order would you do your check? Why?

My thoughts are that you'd check to see if they are different or not, because if they are equal then they won't be consistent or independent, they'll be inconsistent or dependent. I'm not sure which order I would check them though. I'm actually not sure if my logic is even correct :/

Respuesta :

sqdancefan

Answer:

I'd say you need to be more specific.

Step-by-step explanation:

"Different" doesn't tell you much.

Consider the equations ...

  • y = 2x + 1
  • 4x - 2y = -2

These equations are "different", but they are dependent.

_____

I'd mentally (or actually) put the equations in the same form and compare the coefficients of x and y. If they have different ratios, the system is independent and consistent.

If they have the same ratio, the system will not have a single solution. Whether there is no solution or an infinite number of solutions depends on the constant, which I would examine next.

The system above can be put in the form

  • 2x -y = -1
  • 4x -2y = -2

In both cases, the ratio of the x coefficient to the y coefficient is 2/-1 = 4/-2 = -2. This means the lines are at least parallel, if not identical. The numbers in the second equation are all 2 times the numbers in the first equation, so the equations are dependent, and there are an infinite number of solutions. (Both describe the same line.)

If the second equation were 4x -2y = 1, then the two equations would be describing parallel lines, so they would be called inconsistent.

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