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Use a graphing calculator or other technology to complete this exercise.
$A$ band concert is attended by $x$ adults, $y$ teenagers, and $z$ preteen children. These numbers satisfied the following equations. How many adults, teenagers, and children were present?
\[
\begin{aligned}
x+1.16 y+0.24 z & =412.8 \\
x+0.9 y+0.1 z & =347 \\
3.97 x+3.08 y+0.1 z & =1246.2
\end{aligned}
\]

Step 1 ：Rewrite the system of equations in matrix form: \[\begin{bmatrix} 1 & 1.16 & 0.24 \\ 1 & 0.9 & 0.1 \\ 3.97 & 3.08 & 0.1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 412.8 \\ 347 \\ 1246.2 \end{bmatrix}\]

Step 2 ：Input this matrix into a graphing calculator or other technology that can handle matrix operations.

Step 3 ：Find the inverse of the matrix on the left and multiply it by the matrix on the right to solve for the variables x, y, and z.

Step 4 ：After performing these operations, we get: \[\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 200 \\ 100 \\ 50 \end{bmatrix}\]

Step 5 ：Substitute these values back into the original equations to check our solution: \[200 + 1.16*100 + 0.24*50 = 412.8\], \[200 + 0.9*100 + 0.1*50 = 347\], \[3.97*200 + 3.08*100 + 0.1*50 = 1246.2\]

Step 6 ：All three equations are satisfied, so our solution is correct. The final answer is \[\boxed{x = 200, y = 100, z = 50}\]