System of Linear Equations - NonHomogenous Case

[jenl2881]

I'm in Mathematics for Economics and we are using the text "Mathematical Tools for Economics" by Turkington. The question I have comes from the section on the NonHomogenous Case Ax=b concerning Linear Equations. The question (Exercise 2.3 question 2):

Show that the equations

x + 3y - 2z = 9
3x - 17y + 8z = 49
3x - 4y + z = c

do not have a unique solution. For what values of c will they have (a) no solution, (b) an infinite number of solutions. For (b) obtain the general solution.

I'm not really sure what they are asking me to do here. I'm sure it involves doing row operations on the matrix to get to echelon form but I don't really know where to start.

 

[stapel]

Show that the equations

x + 3y - 2z = 9
3x - 17y + 8z = 49
3x - 4y + z = c

do not have a unique solution. For what values of c will they have (a) no solution, (b) an infinite number of solutions. For (b) obtain the general solution.

When you've solved systems of equations and found them to have "no solution", what sort of form did that third line have? When the solution was "infinite", what sort of form did that third line have? When the solution was one point, what sort of form did the three lines have?

Solve the system, as best you can, into "upper triangular" form. Then apply what you've learned by solving other systems, and see where that could lead you here. If you get stuck, please reply showing all of your steps so far. Thank you!

 

[HallsofIvy]

Start by trying to solve the equations for x, y, and z in terms of c. You will find that, at some point you have "0x= f(c)" (or 0y= f(c) or 0z= f(c)) where the right side is some function of c. There will be NO solution if f(c) is not 0 and an infinite number of solutions if f(c)= 0. In the second case, you can solve for two of the variables (x and y, say) in terms of the third.