Consider the system of linear equations y=ax+4 and y=bx-2, where a and b are real numbers. Are these folowing statements always, sometimes or never true. In the description

Question

1. This system has in finely many solutions.
2. This system has no solution.
3. When a>b, the system has one solution.

Are these always true, sometimes true, or never true?

Answer 1

ax+4 = bx-2

(a-b)x = -6

if a=b, there are no real solutions

as =y0+4 never = y=0-2, unless x = infinity, a surreal solution

4=-2 is never true

if ax+4=bx-2 there is one unique real solution

(a-b)x can = -6 when a does not = b

graph them,

when a=b, they are two parallel lines which never intersect

but if a does not =b the lines have different slope and intersect one time

if a>b, then one solution

if a<b then one solution

if a=b then zero solutions

the only way to get an infinite number of solutions is if the two lines are identical, with same slope and same y intercept, but 4 is the y intercept of one line and -2 is the slope of the 2nd line.

they aren't idential lines if they have different y intercepts

There is never an infinite number of real solutions

But there could be an infinite number of surreal solutions, with x and y both infinity plus a finite number