Mathematics

Problem

A fruit vendor sells three types of fruit baskets A, B and C. Basket A contains 2 apples, 1 orange and 3 bananas; basket B contains 1 apple, 3 oranges and 2 bananas; basket C contains 4 apples, 2 oranges and 1 banana. On a certain day the vendor sold 34 apples, 26 oranges and 29 bananas in total. If x, y and z denote the numbers of baskets A, B and C sold respectively, which of the following is the correct matrix equation representing the situation? a)

\begin{pmatrix}2 & 4 & 1 \\ 1 & 2 & 3 \\ 3 & 1 & 2\end{pmatrix}\!\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}34\\26\\29\end{pmatrix}

\begin{pmatrix}2 & 1 & 4 \\ 1 & 3 & 2 \\ 3 & 2 & 1\end{pmatrix}\!\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}29\\34\\26\end{pmatrix}

\begin{pmatrix}2 & 1 & 4 \\ 1 & 3 & 2 \\ 3 & 2 & 1\end{pmatrix}\!\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}34\\26\\29\end{pmatrix}

\begin{pmatrix}2 & 1 & 3 \\ 1 & 3 & 2 \\ 4 & 2 & 1\end{pmatrix}\!\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}34\\26\\29\end{pmatrix}

Intuitive explanation

This is a system of linear equations problem disguised as a word problem. The key insight is to organize the fruit counts by type: each basket type contributes a fixed number of each fruit, and you need to match the total quantities sold. Set up one equation per fruit type (apples, oranges, bananas), where each equation sums the contributions from all three basket types. Then express this system in matrix form $M \mathbf{x} = \mathbf{b}$ , where the coefficient matrix captures how many of each fruit are in each basket.

Intuitive explanation

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