Question

1. mai and tyler are selling items to earn money for their elementary school. the school earns w dollars for every wreath sold and p dollars for every potted plant sold. mai sells 14 wreaths and 3 potted plants and the school earns $70.50. tyler sells 10 wreaths and 7 potted plants and the school earns $62.50. this situation is represented by this system of equations: {14w + 3p = 70.50, 10w + 7p = 62.50}. explain why it makes sense in this situation that the solution of this system is also a solution to 4w+( - 4p)=8.00.

Explanation:

Step1: Write the system of equations

We have the system \(

$$\begin{cases}14w + 3p=70.50\\10w+7p = 62.50\end{cases}$$

\). First, we want to eliminate one of the variables. Let's multiply the first - equation by 5 and the second - equation by 7 to make the coefficients of \(w\) the same in both equations for elimination.
The first equation \(14w + 3p=70.50\) multiplied by 5 gives \(70w+15p = 352.5\).
The second equation \(10w + 7p=62.50\) multiplied by 7 gives \(70w+49p = 437.5\).

Step2: Subtract the first new - equation from the second new - equation

\((70w + 49p)-(70w + 15p)=437.5 - 352.5\).
This simplifies to \(70w+49p - 70w - 15p=85\), then \(34p = 85\).
Solve for \(p\): \(p=\frac{85}{34}=2.5\).

Step3: Substitute \(p = 2.5\) into the first original equation

Substitute \(p = 2.5\) into \(14w+3p = 70.50\). We get \(14w+3\times2.5 = 70.50\).
Which is \(14w+7.5 = 70.50\).
Subtract 7.5 from both sides: \(14w=70.50 - 7.5=63\).
Solve for \(w\): \(w=\frac{63}{14}=4.5\).

Step4: Analyze the given expression \(4w+(-4p)\)

Substitute \(w = 4.5\) and \(p = 2.5\) into \(4w+(-4p)=4w-4p\).
\(4w-4p=4(w - p)=4(4.5 - 2.5)\).
\(4\times2=8\). The value of \(4w+(-4p)\) makes sense in the context of the problem because \(w - p\) represents the difference in the unit - price of wreaths and potted plants, and multiplying by 4 could represent a comparison or calculation related to a quantity of 4 of the difference in earnings per item.

Answer:

The value of \(4w+(-4p)\) is 8. It represents 4 times the difference between the price of a wreath and the price of a potted plant. In the context of the problem, it could be related to a comparison of earnings or costs when considering 4 units of the price - difference between wreaths and potted plants.