2003 AMC 10A Problems/Problem 13
Problem
The sum of three numbers is . The first is four times the sum of the other two. The second is seven times the third. What is the product of all three?
Solution
Let the numbers be , , and in that order.
Therefore, the product of all three numbers is
Alternatively, we can set up the system in matrix form:
is equivalent to
Or, in matrix form
\begin{bmatrix} 1 & 1 & 1 \\ 1 & -4 & -4 \\ 0 & 1 & -7 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix}
=
\begin{bmatrix} 20 \\ 0 \\ 0 \\ \end{bmatrix}
To solve this matrix equation, we can rearrange it thus:
\begin{bmatrix} x \\ y \\ z \\ \end{bmatrix}
=
\begin{bmatrix} 1 & 1 & 1 \\ 1 & -4 & -4 \\ 0 & 1 & -7 \end{bmatrix}
-1
\begin{bmatrix} 20 \\ 0 \\ 0 \\ \end{bmatrix}
Solving this matrix equation by using inverse matrices and matrix multiplication yields
\begin{bmatrix} x \\ y \\ z \\ \end{bmatrix}
=
\begin{bmatrix} 0.5 \\ 3.5 \\ 16 \\ \end{bmatrix}
Which means that x = 0.5, y = 3.5, and z = 16. Therefore, xyz = (0.5)(3.5)(16) = 28