We are given the following system of equations:
[tex]\begin{gathered} 3x+5y=11,(1) \\ 2x+5y=4,(2) \end{gathered}[/tex]We can solve this system of equations using the method of elimination. To do that we will multiply equation (2) by -1:
[tex]-2x-5y=-4,(3)[/tex]Now we add equations (1) and (3):
[tex]3x+5y-2x-5y=11-4[/tex]Adding like terms:
[tex]x=7[/tex]Now we replace the value of "x" is equation (1):
[tex]\begin{gathered} 3(7)+5y=11 \\ 21+5y=11 \end{gathered}[/tex]Now we subtract 21 to both sides:
[tex]\begin{gathered} 5y=11-21 \\ 5y=-10 \end{gathered}[/tex]Dividing both sides by 5:
[tex]\begin{gathered} y=-\frac{10}{5} \\ y=-2 \end{gathered}[/tex]Therefore, the solution of the system is:
[tex](x,y)=(7,-2)[/tex]For the second system of equations:
[tex]\begin{gathered} y=3x+7,(1) \\ y=3x+4,(2) \end{gathered}[/tex]These equations represent two lines with the same slope, and therefore, parallel lines. Since they are parallel lines this means that the system has no solutions.
