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
