Question Type 7: Representing a system of linear equations using matrix notation Exercises

Express the system

Given the system

Represent the system of equations

in the form $AX = B$, clearly defining each matrix.

Given the matrix equation $\begin{pmatrix}3 & 0 & 1\\1 & 2 & -1\\4 & -1 & 2\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix} = \begin{pmatrix}7\\3\\10\end{pmatrix},$ write out each of the three scalar equations.

A juice requires mixing apple, banana, and cherry concentrates. If $x,y,z$ are volumes (litres) of each concentrate, the recipe demands:

A logistics firm packs small ($s$), medium ($m$), and large ($l$) boxes. Total weight and volume constraints give: $5s + 10m + 20l = 200\\2s + 4m + 5l = 50$ Express these two equations in matrix form $AX = B$, where $X = (s,m,l)^T$.

Given the augmented matrix $\begin{pmatrix}1 & -2 & 3 & 5\\0 & 4 & -1 & 2\\2 & 0 & 1 & 6\end{pmatrix},$ write the corresponding system of equations in $x,y,z$.

Let $A = \begin{pmatrix}1 & 2 & 0\\0 & 3 & 4\end{pmatrix},\quad X = \begin{pmatrix}u\\v\\w\end{pmatrix},\quad B = \begin{pmatrix}5\\11\end{pmatrix}.$ Write explicitly the system of linear equations represented by $AX = B$.

A chemical mix requires reagents $A,B,C$ in amounts $x,y,z$ grams satisfying:

An investor allocates funds $x,y,z$ (thousands £) into three projects. Returns satisfy:

Write the system of four equations $x + y + z + w = 6\\2x - y + w = 1\\-x + 3y + 2z = 4\\x + 2z - w = 0$ in the matrix form $AX = B$.