# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 21

Exercises

### Exercise

${\displaystyle {}M,P,S}$ and ${\displaystyle {}T}$ are the members of one family. In this case, ${\displaystyle {}M}$ is three times as old as ${\displaystyle {}S}$ and ${\displaystyle {}T}$ together, ${\displaystyle {}M}$ is older than ${\displaystyle {}P}$, and ${\displaystyle {}S}$ is older than ${\displaystyle {}T}$, moreover, the age difference between ${\displaystyle {}S}$ and ${\displaystyle {}T}$ is twice as large as the difference between ${\displaystyle {}M}$ and ${\displaystyle {}P}$. Furthermore, ${\displaystyle {}P}$ is seven times as old as ${\displaystyle {}T}$, and the sum of the ages of all family members is equal to the paternal grandmother's age, that is ${\displaystyle {}83}$.

a) Set up a linear system of equations that expresses the conditions described.

b) Solve this system of equations.

### Exercise *

Kevin pays ${\displaystyle {}2.50}$€ for a winter bunch of flowers with ${\displaystyle {}3}$ snowdrops and ${\displaystyle {}4}$ mistletoes, and Jennifer pays ${\displaystyle {}2.30}$€ for a bunch with ${\displaystyle {}5}$ snowdrops and ${\displaystyle {}2}$ mistletoes. How much does a bunch with one snowdrop and ${\displaystyle {}11}$ mistletoes cost?

### Exercise

Show that the system of linear equations

${\displaystyle {\begin{matrix}-4x+6y&=&0\\5x+8y&=&0\,\end{matrix}}}$

has only the trivial solution ${\displaystyle {}(0,0)}$.

### Exercise

We look at a clock with hour and minute hands. Now it is 6 o'clock, so that both hands have opposite directions. When will the hands have opposite directions again?

### Exercise

Solve the linear equation

${\displaystyle {}x+y+z=0\,.}$

### Exercise

Solve the system of linear equations

${\displaystyle x=5,\,2y=3,\,4z+w=3.}$

### Exercise

Solve the following system of inhomogeneous linear equations.

${\displaystyle {\begin{matrix}3x&\,\,\,\,\,\,\,\,&+z&+4w&=&4\\2x&+2y&\,\,\,\,\,\,\,\,&+w&=&0\\4x&+6y&\,\,\,\,\,\,\,\,&+w&=&2\\x&+3y&+5z&\,\,\,\,\,\,\,\,&=&3\,.\end{matrix}}}$

### Exercise

Does there exist a solution ${\displaystyle {}(a,b,c)\in \mathbb {Q} ^{3}}$ for the system of linear equations

${\displaystyle {}a{\begin{pmatrix}1\\2\\11\\2\end{pmatrix}}+b{\begin{pmatrix}2\\2\\12\\3\end{pmatrix}}+c{\begin{pmatrix}3\\1\\20\\7\end{pmatrix}}={\begin{pmatrix}1\\2\\20\\5\end{pmatrix}}\,}$

from Example 21.1 ?

### Exercise

Show that for every system of linear equations over ${\displaystyle {}\mathbb {Q} }$, there exists an equivalent linear system with the property that all coefficients are integers.

### Exercise

Bring the system of linear equations

${\displaystyle {}3x-4+5y=8z+7x\,,}$
${\displaystyle {}2-4x+z=2y+3x+6\,,}$
${\displaystyle {}4z-3x+2x+3=5x-11y+2z-8\,}$

into a standard form, and solve it.

### Exercise *

Exhibit a linear equation for the straight line in ${\displaystyle {}\mathbb {R} ^{2}}$, which runs through the two points ${\displaystyle {}(2,3)}$ and ${\displaystyle {}(5,-7)}$.

Before dealing with the next exercise, we recall the concept of a secant, which occurred already in the context of differential calculus.

For a function

${\displaystyle f\colon T\longrightarrow \mathbb {R} }$

defined on a subset ${\displaystyle {}T\subseteq \mathbb {R} }$ and two distinct points ${\displaystyle {}a,b\in T}$, the line through ${\displaystyle {}(a,f(a))}$ and ${\displaystyle {}(b,f(b))}$ is called the secant of ${\displaystyle {}f}$ at

${\displaystyle {}a}$ and ${\displaystyle {}b}$.

### Exercise

Determine an equation for the secant of the function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto -x^{3}+x^{2}+2,}$

to the points ${\displaystyle {}3}$ and ${\displaystyle {}4}$.

### Exercise

Determine a linear equation for the plane in ${\displaystyle {}\mathbb {R} ^{3}}$, where the three points

${\displaystyle (1,0,0),\,(0,1,2){\text{ and }}(2,3,4)}$

lie.

### Exercise

Given a complex number

${\displaystyle {}z=a+b{\mathrm {i} }\neq 0\,,}$

find its inverse complex number with the help of a real system of linear equations, with two equations in two variables.

### Exercise

Solve, over the complex numbers, the linear system of equations

${\displaystyle {\begin{matrix}{\mathrm {i} }x&+y&+(2-{\mathrm {i} })z&=&2\\&7y&+2{\mathrm {i} }z&=&-1+3{\mathrm {i} }\\&&(2-5{\mathrm {i} })z&=&1\,.\end{matrix}}}$

### Exercise

Let ${\displaystyle {}K}$ be the field with two elements. Solve in ${\displaystyle {}K}$ the inhomogeneous linear system

${\displaystyle {\begin{matrix}x&+y&&=&1\\&y&+z&=&0\\x&+y&+z&=&0\,.\end{matrix}}}$

### Exercise

Show with an example that the linear system given by three equations I, II, III is not equivalent to the linear system given by the three equations I-II, I-III, II-III.

The following exercises are also about finding appropriate methods to solve the equations.

### Exercise

Solve the system of linear equations

${\displaystyle {\begin{matrix}&+7y&+3z&\,\,\,\,\,\,\,\,&=&4\\x&\,\,\,\,\,\,\,\,&\,\,\,\,\,\,\,\,&+4w&=&9\\&-3y&-5z&\,\,\,\,\,\,\,\,&=&2\\-2x&\,\,\,\,\,\,\,\,&\,\,\,\,\,\,\,\,&+3w&=&3\,.\end{matrix}}}$

### Exercise

Solve the system of linear equations

${\displaystyle {}7y=5\,,}$
${\displaystyle {}4z=8\,,}$
${\displaystyle {}2u-3v=0\,,}$
${\displaystyle {}5w=0\,,}$
${\displaystyle {}6x-3y+2z-11u-v+5w=17\,,}$
${\displaystyle {}4u-5v=0\,.}$

### Exercise

Solve the system of linear equations

${\displaystyle {}4x-5y+7z=-3\,,}$
${\displaystyle {}-2x+4y+3z=9\,,}$
${\displaystyle {}x=-2\,.}$

### Exercise

Solve the system of linear equations

${\displaystyle {}3x-67y+14z-123u-51w=5\,,}$
${\displaystyle {}8x-11y+12z-27u-65w=51\,,}$
${\displaystyle {}66x-67y-77z-u+100w=0\,,}$
${\displaystyle {}8x-11y+12z-27u-65w=-15\,,}$
${\displaystyle {}-301x+44y+33z-31u-18w=571\,.}$

### Exercise

Determine, in dependence of the parameter ${\displaystyle {}a\in \mathbb {R} }$, the solution space ${\displaystyle {}L_{a}\subseteq \mathbb {R} ^{3}}$ of the system of linear equations

${\displaystyle {}5x+ay+(1-a)z=0\,,}$
${\displaystyle {}2ax+a^{2}y+3z=0\,.}$

### Exercise

A system of linear inequalities is given by

${\displaystyle {}x\geq 0\,,}$
${\displaystyle {}y\geq 0\,,}$
${\displaystyle {}x+y\leq 1\,}$

Sketch the solution set of this system of inequalities.

### Exercise

Let

${\displaystyle {}a_{1}x+b_{1}y\geq c_{1}\,,}$
${\displaystyle {}a_{2}x+b_{2}y\geq c_{2}\,,}$
${\displaystyle {}a_{3}x+b3y\geq c_{3}\,,}$

be a system of linear inequalities, whose solution set is a triangle. How does the solution set look like, when we replace one inequality ${\displaystyle {}\geq }$ by ${\displaystyle {}\leq }$?

Hand-in-exercises

### Exercise (4 marks)

Solve the following system of inhomogeneous linear equations.

${\displaystyle {\begin{matrix}x&+2y&+3z&+4w&=&1\\2x&+3y&+4z&+5w&=&7\\x&\,\,\,\,\,\,\,\,&+z&\,\,\,\,\,\,\,\,&=&9\\x&+5y&+5z&+w&=&0\,.\end{matrix}}}$

### Exercise (3 marks)

Solve the system of linear equations in the variables ${\displaystyle {}x_{1},x_{2},\ldots ,x_{10}}$, which is given by the two equations

${\displaystyle {}x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=0\,}$

and

${\displaystyle {}x_{1}-x_{2}+x_{3}-x_{4}+x_{5}-x_{6}+x_{7}-x_{8}+x_{9}-x_{10}=0\,.}$

### Exercise (3 marks)

Consider in ${\displaystyle {}\mathbb {R} ^{3}}$ the two planes

${\displaystyle E={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 3x+4y+5z=2\right\}}{\text{ and }}F={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 2x-y+3z=-1\right\}}.}$

Determine the intersecting line ${\displaystyle {}E\cap F}$.

### Exercise (3 marks)

Determine a linear equation for the plane in ${\displaystyle {}\mathbb {R} ^{3}}$, where the three points

${\displaystyle (1,0,2),\,\,(4,-3,2)\,{\text{ and }}\,(2,1,-1)}$

lie.

### Exercise (4 marks)

We consider the linear system

${\displaystyle {\begin{matrix}2x&-ay&&=&-2\\ax&&+3z&=&3\\-{\frac {1}{3}}x&+y&+z&=&2\end{matrix}}}$

over the real numbers, depending on the parameter ${\displaystyle {}a\in \mathbb {R} }$. For which ${\displaystyle {}a}$ does the system of equations have no solution, one solution or infinitely many solutions?

### Exercise (4 marks)

Show that a system of linear equations

${\displaystyle {}ax+by=0\,}$
${\displaystyle {}cx+dy=0\,}$

has only the trivial solution ${\displaystyle {}(0,0)}$ if and only if ${\displaystyle {}ad-bc\neq 0}$.

### Exercise (4 (2+2) marks)

A system of linear inequalities is given by

${\displaystyle {}x\geq 0\,,}$
${\displaystyle {}y+x\geq 0\,,}$
${\displaystyle {}-1-y\leq -x\,,}$
${\displaystyle {}5y-2x\geq 3\,.}$

a) Sketch the solution set of this system.

b) Determine the corners for this solution set.