IB Mathematics Applications & Interpretation Question Type 7: Representing a System of Linear Equations Using Matrix Notation Exercises

A chemical mix requires reagents $A,B,C$ in amounts $x,y,z$ grams satisfying: $$\begin{array}{rcl}2x+y-z& =& 4\\ x+3y+2z& =& 7\\ 4x-y+5z& =& 12\end{array}$$

Form the augmented matrix for this system.

An investor allocates funds $x,y,z$ (thousands £) into three projects. Returns satisfy: $$\begin{array}{rl}0.05x+0.02y+0.03z& =4\\ 0.01x+0.06y+0.02z& =3\\ 0.02x+0.01y+0.07z& =5\end{array}$$ Express this system in matrix form and define all matrices.

Express the system $$\{\begin{array}{l}2x-3y=7\\ 4x+y=1\end{array}$$ in matrix form $AX=B$.

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

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

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

Represent the system of equations $$\begin{array}{rl}3x+y+2z& =5\\ 4x+2y+z& =10\\ 2x+y+3z& =9\end{array}$$ in the form $AX=B$, clearly defining each matrix.

Given the system $$\begin{array}{rl}x+2y-z& =4\\ -2x+y+3z& =1\\ 3x-y+2z& =5,\end{array}$$ write it in the form $AX=B$.

Write the system of four equations $$\begin{array}{rl}x+y+z+w& =6\\ 2x-y+w& =1\\ -x+3y+2z& =4\\ x+2z-w& =0\end{array}$$ in the matrix form $AX=B$.

Let $$A=\left(\begin{array}{ccc}1& 2& 0\\ 0& 3& 4\end{array}\right),X=\left(\begin{array}{c}u\\ v\\ w\end{array}\right),B=\left(\begin{array}{c}5\\ 11\end{array}\right).$$ Write explicitly the system of linear equations represented by $AX=B$.

A juice requires mixing apple, banana, and cherry concentrates. If $x,y,z$ are volumes (litres) of each concentrate, the recipe demands: $$\begin{array}{rl}x+2y+z& =10\\ 2x-y+3z& =15\\ 3x+y-z& =5\end{array}$$ Express this system in matrix notation.