Respuesta :
Explanation:
Let x = sculpture
Let y = painting
Let z = photograph
From the statements
The top three prices for works of art sold at auction in 2013 totaled $305.8 million.
and
The selling price of the painting was $51.9 million more than that of the photograph.
[tex]x+y+z=305.8[/tex][tex]y=51.9+z[/tex]Together, the painting and the photograph sold for $25.2 million more than the sculpture.
Thus, we will have
[tex]y+z=x+25.2[/tex]We are to find the value of x, y, and z
We will follow the steps below
step 1:
[tex]\begin{bmatrix}x+y+z=305.8\\ y=51.9+z\\ y+z=x+25.2\end{bmatrix}[/tex]Step 2:
[tex]\begin{gathered} \mathrm{Substitute\:}y=51.9+z \\ \begin{bmatrix}x+51.9+z+z=305.8\\ 51.9+z+z=x+25.2\end{bmatrix} \end{gathered}[/tex]Step 3: Simplify
[tex]\begin{bmatrix}x+51.9+2z=305.8\\ 51.9+2z=x+25.2\end{bmatrix}[/tex]Step 4
[tex]\begin{gathered} \mathrm{Substitute\:}x=-2z+253.9 \\ \begin{bmatrix}51.9+2z=\left(-2z+253.9\right)+25.2\end{bmatrix} \end{gathered}[/tex]Step 5: Simplify
[tex]\begin{bmatrix}51.9+2z=-2z+279.1\end{bmatrix}[/tex]Step 6:
[tex]\begin{gathered} \mathrm{For\:}x=-2z+253.9 \\ \mathrm{Substitute\:}z=56.8 \\ x=-2\times\:56.8+253.9 \\ x=140.3 \end{gathered}[/tex]Step 7
[tex]\begin{gathered} \mathrm{For\:}y=51.9+z \\ \mathrm{Substitute\:}x=140.3,\:z=56.8 \\ y=51.9+56.8 \\ y=108.7 \end{gathered}[/tex][tex]\begin{gathered} \mathrm{The\:solutions\:to\:the\:system\:of\:equations\:are:} \\ x=140.3,\:z=56.8,\:y=108.7 \end{gathered}[/tex]Therefore,
The selling price of the sculpture was $140.3 million.
The selling price of the painting was $108.7 million.
The selling price of the photograph was $56.8 million.
