SOLUTION: Find numbers a and b so that the system of equations
{3x+y = 5
x−ay = b
has (i) no solutions, (ii) infinitely many solutions, and (iii) a unique solution at (1,2). Graph
Question 1079288: Find numbers a and b so that the system of equations
{3x+y = 5
x−ay = b
has (i) no solutions, (ii) infinitely many solutions, and (iii) a unique solution at (1,2). Graph the two lines in all three situations. Answer by ikleyn(52900) (Show Source):
(i) no solutions If a = and b =/= then the system HAS NO solutions.
Indeed, the left sides of equations are proportional with the coefficient 3 (First to the Second),
while the right sides are not proportional with the same coefficient.
The plot is two distinct parallel lines with NO intersection.
(ii) infinitely many solutions If a = and b = then the system HAS INFINITELY NANY solutions.
Indeed, the left sides of equations are proportional with the coefficient 3 (First to the Second),
while the right sides are not proportional with the same coefficient.
Therefore, every solution to the first equation is the solution to the second equation.
Thus the two equations are equivalent to one, either of the two.
The plot is two coinsiding parallel lines with infinitely many common points . . .
(iii) a unique solution at (1,2). If a =/= then the system has a unique solution.
The plot is two non-parallel straight lines that have a unique intersection, which represents
the common solution to the system.
