Step-by-step explanation:
We transform the system of equations to the form:
![\left\{\begin{array}{ccc}ax+by=c\\dx+ey=f\end{array}\right](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bccc%7Dax%2Bby%3Dc%5C%5Cdx%2Bey%3Df%5Cend%7Barray%7D%5Cright)
Where <em>a & b</em> and <em>d & e</em> are relatively prime number.
<h2>1.</h2>
If <em>a ≠ d</em> or <em>b ≠ e</em> then the system of equations has one solution.
<em>Example:</em>
![\left\{\begin{array}{ccc}2x-3y=-4\\3x+3y=9\end{array}\right](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bccc%7D2x-3y%3D-4%5C%5C3x%2B3y%3D9%5Cend%7Barray%7D%5Cright)
<em>Add both sides of equations:</em>
<em>divide both sides by 5</em>
![x=1](https://tex.z-dn.net/?f=x%3D1)
<em>Substitute it to the second equation:</em>
![3(1)+3y=9](https://tex.z-dn.net/?f=3%281%29%2B3y%3D9)
<em>subtract 3 from both sides</em>
<em>divide both sides by 3</em>
![y=2](https://tex.z-dn.net/?f=y%3D2)
![\boxed{x=1,\ y=2\to(1,\ 2)}](https://tex.z-dn.net/?f=%5Cboxed%7Bx%3D1%2C%5C%20y%3D2%5Cto%281%2C%5C%202%29%7D)
<h2>2.</h2>
If <em>a = d</em> and <em>b = e</em> and <em>c = f</em> then the system of equations has infinitely many solutions.
<em>Example:</em>
![\left\{\begin{array}{ccc}2x+3y=5\\2x+3y=5\end{array}\right](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bccc%7D2x%2B3y%3D5%5C%5C2x%2B3y%3D5%5Cend%7Barray%7D%5Cright)
<em>Change the signs in the second equation. Next add both sides of equations:</em>
![\underline{+\left\{\begin{array}{ccc}2x+3y=5\\-2x-3y=-5\end{array}\right}\\.\qquad0=0\qquad\bold{TRUE}](https://tex.z-dn.net/?f=%5Cunderline%7B%2B%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bccc%7D2x%2B3y%3D5%5C%5C-2x-3y%3D-5%5Cend%7Barray%7D%5Cright%7D%5C%5C.%5Cqquad0%3D0%5Cqquad%5Cbold%7BTRUE%7D)
![\boxed{x\in\mathbb{R},\ y=\dfrac{5-2x}{3}}](https://tex.z-dn.net/?f=%5Cboxed%7Bx%5Cin%5Cmathbb%7BR%7D%2C%5C%20y%3D%5Cdfrac%7B5-2x%7D%7B3%7D%7D)
<h2>3.</h2>
If <em>a = d</em> and <em>b = e</em> and <em>c ≠ f</em> then the system of equations has no solution.
<em>Example:</em>
![\left\{\begin{array}{ccc}3x+2y=6\\3x+2y=1\end{array}\right](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bccc%7D3x%2B2y%3D6%5C%5C3x%2B2y%3D1%5Cend%7Barray%7D%5Cright)
<em>Change the signs in the second equation. Next add both sides of equations:</em>
![\underline{+\left\{\begin{array}{ccc}3x+2y=6\\-3x-2y=-1\end{array}\right}\\.\qquad0=5\qquad\bold{FALSE}](https://tex.z-dn.net/?f=%5Cunderline%7B%2B%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bccc%7D3x%2B2y%3D6%5C%5C-3x-2y%3D-1%5Cend%7Barray%7D%5Cright%7D%5C%5C.%5Cqquad0%3D5%5Cqquad%5Cbold%7BFALSE%7D)