I'm not sure what this question is asking:
For which a, b does the following system:
3x - y= b
7x + ay= 1
3x - y=b
have
1) a unique solution
2)infinite solutions
3)no solutions
unique solution means there is one intersection for the two lines. so there is a single x = ? and y = ? that solves the system. as i said, we have a unique solution when the two straight lines intersect
we have infinite solutions when the lines are, in fact, the same line. so one line lies on top of the other, so there are an infinite amount of intersecting points
we have no solution if the lines never intersect, that is, if the lines are parallel.
all you have to do, if figure out the values for a and b that cause the respective situations to happen
Awesome!
Just one more question:
What method should I use to solve it? I have a method in my notes but it doesn't contain any unknowns a or b such as in this problem. Here's what's in my notes:
ax+by=c
1) 2x-3y=5 ---> Multiply by 4 (to get a common x-value)
2) 8x+y=2
Subtract 2) from 1)
13y=-18
y=-13/18
sub y into equation 1) and solve for x
Is this the same thing or am I way off?
i don't think so
here are some hints:
we have:
3x - y = b
7x + ay = 1
to have infinitely many solutions, we need these to be the same line. lets start by getting the x's the same. multiply the first equation by 7/3. then your new equations are
7x - (7/3)y = (7/3)b
7x + ay = 1
the 7x's are the same, so to make sure we have the same line, we need the y's to be the same, and the right hand to be the same. thus we want ay = -(7/3)y and 1 = (7/3)b. from those you can find a and b
for no solution, we want the lines to be parallel. so get both equations in the slope-intercept form of the equation of a line, and make sure you solve for a and b such that the intercepts are different and one slope is the negative inverse of the other
for a unique solution, just make sure neither of the above occur...
good luck. if you need more help, don't be afraid to ask
So,
For infintely many solutions I got:
7x-7/3y=7/3b
7x+ ay=1
ay=-7/3y, a=-7/3
1=7/3b, b=3/7
For no solutions I got:
y=3x-b
y=$\displaystyle 1-7x/a$
To solve for a and b, do I use the same method as before?
For a unique solution, I'm not sure how to do this.
yes
yes, you want the slopes to be the same. you know what the slopes of the two lines are, right? then, make sure to choose b so that the y-intercepts are different.For no solutions I got:
y=3x-b
y=$\displaystyle 1-7x/a$
To solve for a and b, do I use the same method as before?
you can choose b so that the y-intercepts are the same, so they meet at the y-axis, but choose a value of a that makes the slopes differentFor a unique solution, I'm not sure how to do this.