Question

What can you figure out about the prices of the items (hat, pikachu, briefcase)? Show or explain your thinking.( theres another picture to it)

Answer 1

To determine the price of the items on the black corner (4 hats, 7 Pikachu, 2 briefcases), first, you have to determine the price of each item, using the given combinations.

If:

"x" represents the price of one hat

"y" represents the price of one Pikachu doll

"z" represents the price of one briefcase

You can express the given combinations as follows:

Purple corner:

5 hats =5x

1 Pikachu = y

3 briefcases= 3z

`5x+y+3z=237`

Green corner:

8 hats= 8x

3 pikachu= 3y

1 briefcase= z

`8x+3y+z=320`

Red corner:

7 hats= 7x

7 briefcases= 7z

`7x+7z=371`

Once you have expressed all combinations as equations you can start working on calculating the item prices.

The first step is to multiply the first expression by 3:

`\begin{gathered} 3(5x+y+3z)=3\cdot(237) \\ 3\cdot5x+3\cdot y+3\cdot3z=3\cdot237 \\ 15x+3y+9z=711 \end{gathered}`

The second step is to subtract the equation obtained on the first step and second equation:

The result is 7x+8z=391

The third step is to subtract the result on step 2 and the third equation:

The result is z=20

Once we have determined the value of one of the variables, you can determine the other two:

First, replace the value of z into the third equation:

`\begin{gathered} 7x+7z=371 \\ 7x+7\cdot20=371 \\ 7x+140=371 \end{gathered}`

From this expression, you can determine the value of x, pass 140 to the right side of the equal sign by applying the opposite operation to both sides of it:

`\begin{gathered} 7x+140-140=371-140 \\ 7x=231 \end{gathered}`

Next, divide both sides by 7 to determine the value of x:

`\begin{gathered} (7x)/(7)=(231)/(7) \\ x=33 \end{gathered}`

Now that we know the values of x and z, we can determine the value of y by replacing them in one of the equations. I'm going to use the first equation to determine the value of y:

`\begin{gathered} 5x+y+3z=237 \\ 5\cdot33+y+3\cdot20=237 \\ 165+y+60=237 \\ 225+y=237 \end{gathered}`

Pass 225 to the right side of the equal sign by applying the opposite operation to both sides of it:

`\begin{gathered} 225-225+y=237-225 \\ y=12 \end{gathered}`

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