Question

point(s) possibleThe top three prices for works of art sold at auction in 2013 totaled $305.8 million. These three works of art were a sculpture, a painting, and a photograph. The sellingprice of the painting was $51.9 million more than that of the photograph. Together, the painting and the photograph sold for $25.2 million more than the sculpture.What was the selling price of each work?The selling price of the sculpture was $ million.(Type an integer or a decimal. Do not round.)The selling price of the painting was $ million.(Type an integer or a decimal. Do not round.)The selling price of the photograph was $2 million(Type an integer or a decimal. Do not round.)

Answer 1

Step-by-step 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.

`x+y+z=305.8`
`y=51.9+z`

Together, the painting and the photograph sold for $25.2 million more than the sculpture.

Thus, we will have

`y+z=x+25.2`

We are to find the value of x, y, and z

We will follow the steps below

step 1:

`\begin{bmatrix}x+y+z=305.8\\ y=51.9+z\\ y+z=x+25.2\end{bmatrix}`

Step 2:

`\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}`

Step 3: Simplify

`\begin{bmatrix}x+51.9+2z=305.8\\ 51.9+2z=x+25.2\end{bmatrix}`

Step 4

`\begin{gathered} \mathrm{Substitute\:}x=-2z+253.9 \\ \begin{bmatrix}51.9+2z=\left(-2z+253.9\right)+25.2\end{bmatrix} \end{gathered}`

Step 5: Simplify

`\begin{bmatrix}51.9+2z=-2z+279.1\end{bmatrix}`

Step 6:

`\begin{gathered} \mathrm{For\:}x=-2z+253.9 \\ \mathrm{Substitute\:}z=56.8 \\ x=-2*\:56.8+253.9 \\ x=140.3 \end{gathered}`

Step 7

`\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}`
`\begin{gathered} \mathrm{The\:solutions\:to\:the\:system\:of\:equations\:are:} \\ x=140.3,\:z=56.8,\:y=108.7 \end{gathered}`

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.