两个商店的面条套餐和米饭套餐的价格如下所示。

((a))
计算 $\displaystyle \begin{pmatrix}2.5 & 3.5\\[4pt]2.8 & 3.2\end{pmatrix}\begin{pmatrix}2\\[4pt]1\end{pmatrix}$ 。

答案 ___
((b))
描述你在 (a) 中的答案代表什么。
((c))
莎莉必须从某一家商店购买总共 6 份套餐。
她应该买多少份面条套餐和米饭套餐，才能使无论她是在 R 商店还是 S 商店购买，总费用都相同？

答案 ___ 份面条套餐，___ 份米饭套餐

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完整解法

22((a))

1. Multiply each row of the matrix by the column vector.

\begin{pmatrix}2.5 & 3.5\\2.8 & 3.2\end{pmatrix}\begin{pmatrix}2\\1\end{pmatrix} = \begin{pmatrix}2.5(2)+3.5(1)\\2.8(2)+3.2(1)\end{pmatrix}

2. Evaluate the two entries of the resulting column vector.

\begin{pmatrix}2.5(2)+3.5(1)\\2.8(2)+3.2(1)\end{pmatrix} = \begin{pmatrix}5+3.5\\5.6+3.2\end{pmatrix} = \begin{pmatrix}8.5\\8.8\end{pmatrix}

答案

$\begin{pmatrix}8.5\\8.8\end{pmatrix}$

22((b))

1. Identify what the column vector represents.

The column vector $\begin{pmatrix}2\\1\end{pmatrix}$ means $2$ noodle sets and $1$ rice set.

2. Interpret the two entries of the answer from part (a).

From part (a), $\begin{pmatrix}8.5\\8.8\end{pmatrix}$ corresponds to the total costs at the two shops. The first entry is for Shop R and the second entry is for Shop S.

答案

The answer represents the total cost of buying $2$ noodle sets and $1$ rice set from each shop: $8.50 from Shop R and $8.80 from Shop S.

22((c))

1. Define variables and express the total cost at each shop.

Let the number of noodle sets be $x$ .

Since Sally buys a total of $6$ meal sets, the number of rice sets is $6-x$ .

Cost at Shop R:
$2.50x+3.50(6-x)$
Cost at Shop S:
$2.80x+3.20(6-x)$

2. Equate the two costs and solve for the number of noodle and rice sets.

For the total cost to be the same,
$2.50x+3.50(6-x)=2.80x+3.20(6-x)$
$2.50x+21-3.50x=2.80x+19.20-3.20x$
$21-x=19.20-0.40x$
$1.80=0.60x$
$x=3$

So Sally buys $3$ noodle sets and $6-3=3$ rice sets.

答案

$4$ noodle set(s), $2$ rice set(s)

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两个商店的面条套餐和米饭套餐的价格如下所示。

完整解法

22((a))

22((b))

22((c))

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