I commented on your question but I'll solve it as if the second scenario had 4 hot dogs and 3 hamburgers totaling $13.75
Let D represent hot dogs. Let H represent hamburgers.
[tex]3D+4H=14.25\\ 4D+3H=13.75[/tex]
This is called a system of equations. You must substitute one equation into the other. I'll work it through, and hopefully you can follow along.
[tex]3D+4H=14.25\\ 3D+4H-4H=14.25-4H\\ 3D=14.25-4H\\ \frac{3D}{3}=\frac{14.25}{3}-\frac{4H}{3}\\ D=\frac{19}{4}-\frac{4}{3}H[/tex]
Now you have the value of one hot dog (D). Substitute this value into the other equation. This way you will only be working with the H variable.
[tex]4D+3H=13.75\\ 4(\frac{19}{4}-\frac{4}{3}H)+3H=13.75\\ 19-\frac{16}{3}H+3H=13.75\\ -\frac{16}{3}H+3H=13.75-19\\ -\frac{7}{3}H=-\frac{21}{4}\\ -\frac{7}{3}H*\frac{3}{7}=-\frac{21}{4}*\frac{3}{7}\\ -1H=-\frac{9}{4}\\ H=\frac{9}{4}[/tex]
9/4=2.25 for the price of a Hamburger (H).
Now plug the value for H (2.25) into either equation.
[tex]3D+4H=14.25\\ 3D+4(2.25)=14.25\\ 3D+9=14.25\\ 3D=5.25\\ D=1.75[/tex]
The price for a Hamburger is $2.25
The price for a Hot Dog is $1.75
