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

Exercises

Exercise Create referencenumber

${}M,P,S$ 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 * Create referencenumber

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?

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Show that the system of linear equations

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

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

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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?

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Solve the linear equationMDLD/linear equation

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

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Solve the system of linear equations

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

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Solve the following system of inhomogeneous linear equations.

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

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

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

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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 * Create referencenumber

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

${}a$ and ${}b$.

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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 ${}4$.

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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.

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Given a complex numberMDLD/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.

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Solve, over the complex numbers,MDLD/complex numbers the linear systemMDLD/linear system of equations

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

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

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

Exercise Create referencenumber

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.

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Solve the system of linear equations

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

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Solve the system of linear equations

${}7y=5\,,$

${}4z=8\,,$

${}2u-3v=0\,,$

${}5w=0\,,$

${}6x-3y+2z-11u-v+5w=17\,,$

${}4u-5v=0\,.$

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Solve the system of linear equations

${}4x-5y+7z=-3\,,$

${}-2x+4y+3z=9\,,$

${}x=-2\,.$

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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 Create referencenumber

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 Create referencenumber

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.

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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) Create referencenumber

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) Create referencenumber

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) Create referencenumber

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) Create referencenumber

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) Create referencenumber

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) Create referencenumber

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) Create referencenumber

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.

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