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

    Exercises

### Exercise

$M,P,S$ and $T$ are the members of one family. In this case, $M$ is three times as old as $S$ and $T$ together, $M$ is older than $P$, and $S$ is older than $T$, moreover, the age difference between $S$ and $T$ is twice as large as the difference between $M$ and $P$. Furthermore, $P$ is seven times as old as $T$, and the sum of the ages of all family members is equal to the paternal grandmother's age, that is $83$.
a) Set up a linear system of equations that expresses the conditions described.
b) Solve this system of equations.

### Exercise *

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

### Exercise

Show that the system of linear equations
    ${\begin{matrix}-4x+6y&=&0\\5x+8y&=&0\,\end{matrix}}$
has only the trivial solution $(0,0)$.

### 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
    $x+y+z=0\,.$

### Exercise

Solve the system of linear equations
    $x=5,\,2y=3,\,4z+w=3.$

### Exercise

Solve the following system of inhomogeneous linear equations.
    ${\begin{matrix}3x&\,\,\,\,\,\,\,\,&+z&+4w&=&4\\2x&+2y&\,\,\,\,\,\,\,\,&+w&=&0\\4x&+6y&\,\,\,\,\,\,\,\,&+w&=&2\\x&+3y&+5z&\,\,\,\,\,\,\,\,&=&3\,.\end{matrix}}$

### Exercise

Does there exist a solution $(a,b,c)\in \mathbb {Q} ^{3}$ for the system of linear equations
    $a{\begin{pmatrix}1\\2\\11\\2\end{pmatrix}}+b{\begin{pmatrix}2\\2\\12\\3\end{pmatrix}}+c{\begin{pmatrix}3\\1\\20\\7\end{pmatrix}}={\begin{pmatrix}1\\2\\20\\5\end{pmatrix}}\,$
from Example 21.1 ?

### Exercise

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

### Exercise

Bring the system of linear equations
    $3x-4+5y=8z+7x\,,$
    $2-4x+z=2y+3x+6\,,$
    $4z-3x+2x+3=5x-11y+2z-8\,$
into a standard form, and solve it.

### Exercise *

Exhibit a linear equation for the straight line in $\mathbb {R} ^{2}$, which runs through the two points $(2,3)$ and $(5,-7)$.
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
    $f\colon T\longrightarrow \mathbb {R}$
defined on a subset $T\subseteq \mathbb {R}$ and two distinct points $a,b\in T$, the line through $(a,f(a))$ and $(b,f(b))$ is called the secant of $f$ at
$a$
$b$

### Exercise

Determine an equation for the secant of the function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto -x^{3}+x^{2}+2,$
to the points $3$ and $4$.

### Exercise

Determine a linear equation for the plane in $\mathbb {R} ^{3}$, where the three points
    $(1,0,0),\,(0,1,2){\text{ and }}(2,3,4)$
lie.

### Exercise

Given a
complex number
    $z=a+b{\mathrm {i} }\neq 0\,,$
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
    ${\begin{matrix}{\mathrm {i} }x&+y&+(2-{\mathrm {i} })z&=&2\\&7y&+2{\mathrm {i} }z&=&-1+3{\mathrm {i} }\\&&(2-5{\mathrm {i} })z&=&1\,.\end{matrix}}$

### Exercise

Let $K$ be the field with two elements. Solve in $K$ the inhomogeneous linear system
    ${\begin{matrix}x&+y&&=&1\\&y&+z&=&0\\x&+y&+z&=&0\,.\end{matrix}}$

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

### Exercise

Solve the system of linear equations
    $7y=5\,,$
    $4z=8\,,$
    $2u-3v=0\,,$
    $5w=0\,,$
    $6x-3y+2z-11u-v+5w=17\,,$
    $4u-5v=0\,.$

### Exercise

Solve the system of linear equations
    $4x-5y+7z=-3\,,$
    $-2x+4y+3z=9\,,$
    $x=-2\,.$

### Exercise

Solve the system of linear equations
    $3x-67y+14z-123u-51w=5\,,$
    $8x-11y+12z-27u-65w=51\,,$
    $66x-67y-77z-u+100w=0\,,$
    $8x-11y+12z-27u-65w=-15\,,$
    $-301x+44y+33z-31u-18w=571\,.$

### Exercise

Determine, in dependence of the parameter $a\in \mathbb {R}$, the solution space $L_{a}\subseteq \mathbb {R} ^{3}$ of the system of linear equations
    $5x+ay+(1-a)z=0\,,$
    $2ax+a^{2}y+3z=0\,.$

### Exercise

A system of linear inequalities is given by
    $x\geq 0\,,$
    $y\geq 0\,,$
    $x+y\leq 1\,$
Sketch the solution set of this system of inequalities.

### Exercise

Let
    $a_{1}x+b_{1}y\geq c_{1}\,,$
    $a_{2}x+b_{2}y\geq c_{2}\,,$
    $a_{3}x+b3y\geq c_{3}\,,$
be a system of linear inequalities, whose solution set is a triangle. How does the solution set look like, when we replace one inequality $\geq$ by $\leq$?
    Hand-in-exercises

### Exercise (4 marks)

Solve the following system of inhomogeneous linear equations.
    ${\begin{matrix}x&+2y&+3z&+4w&=&1\\2x&+3y&+4z&+5w&=&7\\x&\,\,\,\,\,\,\,\,&+z&\,\,\,\,\,\,\,\,&=&9\\x&+5y&+5z&+w&=&0\,.\end{matrix}}$

### Exercise (3 marks)

Solve the system of linear equations in the variables $x_{1},x_{2},\ldots ,x_{10}$, which is given by the two equations
    $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=0\,$
and
    $x_{1}-x_{2}+x_{3}-x_{4}+x_{5}-x_{6}+x_{7}-x_{8}+x_{9}-x_{10}=0\,.$

### Exercise (3 marks)

Consider in $\mathbb {R} ^{3}$ the two planes
    $E={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 3x+4y+5z=2\right\}}{\text{ and }}F={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 2x-y+3z=-1\right\}}.$
Determine the intersecting line $E\cap F$.

### Exercise (3 marks)

Determine a linear equation for the plane in $\mathbb {R} ^{3}$, where the three points
    $(1,0,2),\,\,(4,-3,2)\,{\text{ and }}\,(2,1,-1)$
lie.

### Exercise (4 marks)

We consider the linear system
    ${\begin{matrix}2x&-ay&&=&-2\\ax&&+3z&=&3\\-{\frac {1}{3}}x&+y&+z&=&2\end{matrix}}$
over the real numbers, depending on the parameter $a\in \mathbb {R}$. For which $a$ does the system of equations have no solution, one solution or infinitely many solutions?

### Exercise (4 marks)

Show that a system of linear equations
    $ax+by=0\,$
    $cx+dy=0\,$
has only the trivial solution $(0,0)$ if and only if $ad-bc\neq 0$.

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

A system of linear inequalities is given by
    $x\geq 0\,,$
    $y+x\geq 0\,,$
    $-1-y\leq -x\,,$
    $5y-2x\geq 3\,.$
a) Sketch the solution set of this system.
b) Determine the corners for this solution set.