How do you know if a system or equations has one solution, no solution, or infinitely many solutions

Step-by-step explanation:We transform the system of equations to the form:[tex]\left\{\begin{array}{ccc}ax+by=c\\dx+ey=f\end{array}\right[/tex]Where a & b and d & e are relatively prime number.1.If a ≠ d or b ≠ e then the system of equations has one solution.Example:[tex]\left\{\begin{array}{ccc}2x-3y=-4\\3x+3y=9\end{array}\right[/tex]Add both sides of equations:[tex]5x=5[/tex]     divide both sides by 5[tex]x=1[/tex]Substitute it to the second equation:[tex]3(1)+3y=9[/tex][tex]3+3y=9[/tex]           subtract 3 from both sides[tex]3y=6[/tex]       divide both sides by 3[tex]y=2[/tex][tex]\boxed{x=1,\ y=2\to(1,\ 2)}[/tex]2.If a = d and b = e and c = f then the system of equations has infinitely many solutions.Example:[tex]\left\{\begin{array}{ccc}2x+3y=5\\2x+3y=5\end{array}\right[/tex]Change the signs in the second equation. Next add both sides of equations:[tex]\underline{+\left\{\begin{array}{ccc}2x+3y=5\\-2x-3y=-5\end{array}\right}\\.\qquad0=0\qquad\bold{TRUE}[/tex][tex]\boxed{x\in\mathbb{R},\ y=\dfrac{5-2x}{3}}[/tex]3.If  a = d and b = e and c ≠ f then the system of equations has no solution.Example:[tex]\left\{\begin{array}{ccc}3x+2y=6\\3x+2y=1\end{array}\right[/tex]Change the signs in the second equation. Next add both sides of equations:[tex]\underline{+\left\{\begin{array}{ccc}3x+2y=6\\-3x-2y=-1\end{array}\right}\\.\qquad0=5\qquad\bold{FALSE}[/tex]