\(5x+ay=2 \\
bx+3y=8\)
Write it in the y=mx+b form.
\(ay=-5x+2 \\
3y=-bx+8 \\ \\
y=-\frac{5}{a}x+\frac{2}{a} \\
y=-\frac{b}{3}x+\frac{8}{3}\)
The system will be inconsistent when the equations describe two parallel lines.
Two lines are parallel when their slopes are identical.
\(-\frac{5}{a}=-\frac{b}{3} \\
ab=(-5) \times (-3) \\
ab=15\)
Also they can't be the same line, so the y-intercepts must be different.
\(\frac{2}{a}
ot= \frac{8}{3} \\
8a
ot= 2 \times 3 \\
8a
ot= 6 \\
a
ot= \frac{6}{8} \\
a
ot= \frac{3}{4}\)
So for every value of a and b such that ab=15 and a≠3/2 this system of equations is inconsisent.
It can be:
\(5x+5y=2 \\
3x+3y=8\)
or
\(5x+15y=2 \\
x+3y=8\)
or
\(5x+\frac{1}{2}y=2 \\
30x+3y=8\)
etc.
