Determine whether the system is independent, dependent, or inconsistent

Question

Determine whether the system is independent, dependent, or inconsistent. You may solve them or graph them. See page 120 in your textbook when classifying a system.

  

     -3x + y = 1

 

     9x - 3y = -3

Answer 1

The first step is to ask whether you understand what they are asking. What is the question? There are many long words...independent, dependent, or inconsistent. You are given the equations of two lines: -3x+y=1 & 9x-3y=-3. Thinking graphically, there are three cases: 1) Either the two lines can intersect at one point [most common case], 2) the lines could be parallel [have the same slope] but with different y-intercepts, so they never intercept, or 3) the two equations are just different by a common factor, meaning they have the same slope & y-intercept, so describe the same line.

To return to the original question, "independent" means case 1) above, dependent refers to case 3), and inconsistent refers to case 2), meaning there is "no solution" because the two lines never intercept.

For this specific problem, I recognize if one multiplies the first equation by -3, they get the second equation. This is case 3)...Conclusion: the system is dependent.