How do I solve to find values for a and b?

[physics=world]

1.

a and b are constant.

I need to find the values of a and b that will make the system a a)unique solution, b)no solution, and c)infinitely many solution.

4x - 2y = 9
ax + by = 6

I know how to do it for a one variable constant. Say instead of a and b there was a k, but i do not know what steps to take to solve for two variable constant.

Homework Equations

3. I solve for infinitely many solution and i got a = 8/3 and b = 4/3.

but I'm having problems on solving for no solution.
I tried to get the coefficients the same, but you cannot multiply it by any number to make it the same coefficient as the first equation. So, i just got a = 4 b = -2 from the equations. and that seems to work. i was wondering if that is correct to do.


For a unique solution, if i just plug in a = 4 and b = 2 that gives me two intersecting line. but i was wondering how do I actually solve it to get a unique solution. What are the steps?

 

[HallsofIvy]

I suggest you try to solve the equations and see what happens! If you multiply the first equation by b and the second equation by 2 you get 4bx- 2by= 9b and 2ax+ 2by= 12. Adding the two equations gives (4b+2a)x= 9b- 12. Now under what conditions does this have (a) a unique solution, (b)no solution, (c) infinite solutions.

 

[Enigman]

Your solution for 'b' for infinite solutions is wrong...Check the sign
When does a system of equation have no solution?

 

[physics=world]

I thought a system of equations had no solution when they were parallel lines. If b = 2 it comes out as two intersecting lines when i graph it.

 

[physics=world]

@hallsofivy I am not understanding what your doing. Can you explain? How can you tell from this eq. alone
(4b+2a)x= 9b- 12 if it satisfy the conditions?

because the way I solved for infinitely many solution was by multiplying 2 to the first eq. and 3 to the 2nd eq.

and got

8x -4y = 18
3ax - 3by = 18

and from that i got a = 8/3 and b = 4/3

 

[Enigman]

I thought a system of equations had no solution when they were parallel lines. If b = 2 it comes out as two intersecting lines when i graph it.

Indeed, as I said before check b's sign.
Also a line has infinite parallels the one you found is just one of them.

 

[verty]

In this case you could convert each statement to the form y = mx + c, then it is should be very easy to read off the answers.

 

[HallsofIvy]

@hallsofivy I am not understanding what your doing. Can you explain? How can you tell from this eq. alone
(4b+2a)x= 9b- 12 if it satisfy the conditions?

because the way I solved for infinitely many solution was by multiplying 2 to the first eq. and 3 to the 2nd eq.

and got

8x -4y = 18
3ax - 3by = 18

and from that i got a = 8/3 and b = 4/3

You got "a= 8/3 and b= 4/3" to answer which question? You were asked to find conditions on a and b for three separate results.

It seems strange that you would be given a question like this if you did not know how to solve pairs of equations. I multiplied the first equation by b and the second 2 to get "-2by" and "+2by" so that, adding the equations, I eliminated y leaving an equation for x only, (4b+2a)x= 9b- 12. If the coefficient is not 0, we can divide both sides by it, getting a unique solution. If 4b+ 2a= 0 but 9b- 12 is NOT 0, the left side is 0 no matter what x is while the right side is not so there is no solution. Finally, if both 4b+ 2a= 0 and 9b- 12= 0 the equation is "0= 0" which is true for all x so there are an infinite number of solutions.

Of course, 9b- 12= 0 is the same as 9b= 12 or b= 12/9= 4/3 and then 4b+ 2a= 16/2+ 2a= 0 so that 2a= -16/3 and a= -8/3, not "8/3". a= -8/3, b= 4/3 give parallel lines so an infinite numbrer of solutions.