We are told that a theater have sold 6 adult tickets and 3 child tickets for a total of 96. If "x" is the price for adult and "y" the price for children, then we can describe the situation mathematically as:
[tex]6x+3y=96,(1)[/tex]Now we are told that there were sold 8 adult tickets and 2 children tickets for a total of 112. This can be represented mathematically as:
[tex]8x+2y=112,(2)[/tex]This gives us a system of two equation with two variables. To solve the system we will use the method of subtitution. First, we will solve for "y" in equation (1):
[tex]6x+3y=96[/tex]Subtracting 6x to both sides:
[tex]3y=96-6x[/tex]Dividing both sides by 3:
[tex]\begin{gathered} y=\frac{96}{3}-\frac{6}{3}x \\ y=32-2x \end{gathered}[/tex]Then we will replace this value of "y" in equation (2):
[tex]8x+2(32-2x)=112[/tex]Using the distributive property:
[tex]8x+64-4x=112[/tex]Adding like terms:
[tex]4x+64=112[/tex]Subtracting 64 to both sides:
[tex]4x=112-64[/tex]Solving the operation:
[tex]4x=48[/tex]Dividing both sides by 4:
[tex]x=\frac{48}{4}=12[/tex]Therefore, x = 12. Replacing this value in equation (1):
[tex]y=32-2(12)[/tex]Solving the operations:
[tex]y=8[/tex]Therefore, adult tickets cost $12 and children tickets cost $8.