Expert: Sherry Wallin - 12/4/2010

Question
QUESTION: Describe the system 6x-2=10 and 9x-3y=8 as consistent and independent, consistent and dependent, or inconsistent. Explain.

Please help, thanks! :)

ANSWER: Anthony~

Consistent and independent means that there is one and only one solution, i.e., that the  two lines cross only one time. Consistent and dependent means there are an infinite number of solutions which in turn implies that the equations represent the same line. Inconsistent means there is no solution which also means that the two lines are parallel which is why they would never cross. One thing you can do is put both in the form of y = mx + b and compare the slopes. If the slopes are the same and they are not the same line then they are parallel. If they are the same line then there is an infinite number of solutions. Let's try doing that, solve each for y:

6x - 2 = 10  hmmm... you cannot solve for y since there is no y but you can solve for x: 6x = 12 => x = 2 which is the equation of a vertical line

Solve the other for y: 9x - 3y = 8 => -3y = -9x + 8 => y = 3x - 8/3
These are not parallel because the first one has no slope and the 2nd one has slope of 3 so it is impossible for the two lines to have the same slope, so they are neither parallel or the same line. The only other possibility is that they are independent, consistent and that there is one and only one solution. Go ahead and substitute x = 2 into the 2nd equation: 9(2) - 3y = 8 => 18 - 3y = 8 => 10 - 3y =>
y = 10/3 which is the one and only one solution: (2, 10/3)

Math Prof

---------- FOLLOW-UP ----------

QUESTION: wow thanks, the first paragraph was very informative. But about the equation that doesn't have "y" .. I had a typo ..t he equation is 6x-2y=10.. I'm sorry. how does that change the outcome?

Answer
Anthony~

6x -2y = 10  solve for y: -2y = -6x + 10 => y = 3x -5

Recall the other equation in slope intercept form is y = 3x - 8/3 and the two equations are not the same, however, notice that m = slope = 3 is the same, thus these two equations are parallel which in turn means their is no solution and the system is inconsistent. Two parallel lines can never cross so there is not point of intersection thus no solution.

Math Prof