<em>Two families are planning a trip to Disney. The Smith family bought tickets for 2 adults and 3 children for $557. The Jones family bought tickets for 2 adults and 1 child </em><em>for $459</em><em>. How much does and adult and child ticket cost?</em>

To solve this problem, we need to write a system of linear equations in two variables. So, we know some facts:

Two families are planning a trip to Disney.
The Smith family bought tickets for 2 adults and 3 children for $557.
The Jones family bought tickets for 2 adults and 1 child for $459.

Let:

$x:Cost \ of \ ticket \ per \ adult \\ \\ y: Cost \ of \ ticket \ per \ child$

For the Smith family:

Cost for the 2 adults:

$2x$

Cost for the 3 children:

$3y$

Total cost:

$2x+3y=557$

For the Jones family:

Cost for the 2 adults:

$2x$

Cost for the 1 child:

$y$

Total cost:

$2x+y=459$

So we have the following system of linear equations:

$\begin{array}{c}(1)\\(2)\end{array}\left\{ \begin{array}{c}2x+3y=557\\2x+y=459\end{array}\right.$

Subtracting (2) from (1):

$\begin{array}{c}(1)\\(2)\end{array}\left\{ \begin{array}{c}2x+3y=557\\-(2x+y=459)\end{array}\right. \\ \\ \\ \begin{array}{c}(1)\\(2)\end{array}\left\{ \begin{array}{c}2x+3y=557\\-2x-y=-459\end{array}\right. \\ \\ \\ (2x-2x)+(3y-y)=557-459 \\ \\ 2y=98 \\ \\ y=49 \\ \\ \\ Finding \ x \ from \ (1): \\ \\ 2x+3(49)=557 \\ \\ 2x=557-147 \\ \\ 2x=410 \\ \\ x=205$