please help!!?

Question

ive tried multiple times this problem and im trying to figure it still could u please help me?
please help with trigonometry!?
this is from my trig homework:
consider the system : 3x+6y=8
rx+sy=t
find r, s, and t so that there are
a. no solutions
b. infinite solutions
c. one solution (2/3, 1)
pleeeease help im having a lot of trouble

Answer 1

These are both equations of lines. Rewrite each equation in the problem in slope-intercept form:

y = mx + b

Geometrically, the solution of the two equations is the intersection of those lines.

(a)
If two lines are parallel but not coinciding, they don't intersect at all. This means that they have the same slope but different intercepts. Choose (r,s,t) such that the two equations have the same m but different b's.

(b)
If two lines are coinciding, they intersect infinitely many times because they are the same line. Choose (r,s,t) such that each equation has the same slope and the same intercept.

(c)
If two lines have different slopes, there is exactly one intersection point. Choose ANY r and s (these are the only parameters that affect m) such that the slope of the (r,s,t) line is not the same as the other line. Then evaluate rx+sy = t to get the value of t when x=2/3, y = 1.

Answer 2

a. There will be no solutions if the 2 functions have the same slope but different y intercepts. To make a different y intercept, simply change the constant, 8. You can make the 8 anything you'd like.
b. If there are infinite solutions, then the functions are the same and may just be written differently. 3x+6y=8 is the same equation as -6x-12y=-16.
c. If there is one solution, then the graphs cross paths only once. Basically, just find values of r,s, and t so that they work with the equation rx+st=t and arent the same as 3x+6y=8.
For example, 0x + 8y=8 is a suitable equation. It reduces to y=1.

Answer 3

yeah what he said