Expert: Sherry Wallin - 9/5/2010

Question
given the linear system
2x-y=3
ax+by=6
find value of a and b such that the resulting system has
a) a unique solution
b) an infinite number of solutions
c) no solution
graph by any graphing software the lines for each of the three cases.

Answer
unique~

First let me apologize for taking this long to respond I haven't been near a computer but I could see your question on my cell phone.

The next thing is you need to know and be focused on what kind of equations will give you the results you are looking for.

Unique means there is only one solution and two lines that cross at only one point will give you one or a unique solution. To have an infinite number of solutions this means you want the equations to be equivalent, i.e., multiples of each other, and finally to have no solution you want the lines to be parallel because then they never cross when you graph them.

a)  any value of 'a' that is different from 2 will work because then the lines will have different slopes and then they can ONLY be unique with one solution so try 3x + -y =3 -> a = 3  b = -1

Note:  if you don't keep all the other terms the same you would have to solve in slope intercept form and then change the slope


b)Just multiply the first equation by some number other than 1 like 2 and then you get a multiple of the first: 4x + -2y = 6-> a = 4 b = -2

c)  You need to solve the first equation in slope intercept form to find the slope and then just write any linear equation with the same slope:  

2x - y = 3
-2x      -2x
-y = -2x + 3
-y/-1 = (-2x + 3)/-1
y = 2x - 3

Now you know the slope is 2 how about y = 2x +1? where a = 2 and b = 1

Math Prof