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, which 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} - 4 x + 6 y & = & 0 \\ 5 x + 8 y & = & 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, 2 y = 3, 4 z + w = 3.

Exercise

Solve the following system of inhomogeneous linear equations.

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

Exercise

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

a (\begin{matrix} 1 \\ 2 \\ 11 \\ 2 \end{matrix}) + b (\begin{matrix} 2 \\ 2 \\ 12 \\ 3 \end{matrix}) + c (\begin{matrix} 3 \\ 1 \\ 20 \\ 7 \end{matrix}) = (\begin{matrix} 1 \\ 2 \\ 20 \\ 5 \end{matrix})

from Example 21.1 ?

Exercise

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

Exercise

Bring the system of linear equations

3 x - 4 + 5 y = 8 z + 7 x,

2 - 4 x + z = 2 y + 3 x + 6,

4 z - 3 x + 2 x + 3 = 5 x - 11 y + 2 z - 8,

into a standard form, and solve it.

Exercise *

Exhibit a linear equation for the straight line in $ℝ^{2}$ , 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 : T ⟶ ℝ

defined on a subset $T \subseteq ℝ$ 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

.

Exercise

Determine an equation for the secant of the function

ℝ ⟶ ℝ, x ⟼ - x^{3} + x^{2} + 2,

to the points $3$ and $4$ .

Exercise

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

(1, 0, 0), (0, 1, 2), and (2, 3, 4)

lie.

Exercise

Given a complex number

z = a + b 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} i x & + y & + (2 - i) z & = & 2 \\ 7 y & + 2 i z & = & - 1 + 3 i \\ (2 - 5 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 by 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} + 7 y & + 3 z & = & 4 \\ x & + 4 w & = & 9 \\ - 3 y & - 5 z & = & 2 \\ - 2 x & + 3 w & = & 3 . \end{matrix}

Exercise

Solve the system of linear equations

7 y = 5,

4 z = 8,

2 u - 3 v = 0,

5 w = 0,

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

4 u - 5 v = 0 .

Exercise

Solve the system of linear equations

4 x - 5 y + 7 z = - 3,

- 2 x + 4 y + 3 z = 9,

x = - 2 .

Exercise

Solve the system of linear equations

3 x - 67 y + 14 z - 123 u - 51 w = 5,

8 x - 11 y + 12 z - 27 u - 65 w = 51,

66 x - 67 y - 77 z - u + 100 w = 0,

8 x - 11 y + 12 z - 27 u - 65 w = - 15,

- 301 x + 44 y + 33 z - 31 u - 18 w = 571 .

Exercise

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

5 x + a y + (1 - a) z = 0,

2 a x + a^{2} y + 3 z = 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 + b 3 y \geq c_{3},

be a system of linear inequalities, whose solution set is a triangle. How does the solution set look, 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 & + 2 y & + 3 z & + 4 w & = & 1 \\ 2 x & + 3 y & + 4 z & + 5 w & = & 7 \\ x & + z & = & 9 \\ x & + 5 y & + 5 z & + w & = & 0 . \end{matrix}

Exercise (3 marks)

Solve the system of linear equations in the variables $x_{1}, x_{2}, \dots, x_{10}$ , 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 $ℝ^{3}$ the two planes

E = {(x, y, z) \in ℝ^{3} ∣ 3 x + 4 y + 5 z = 2} and F = {(x, y, z) \in ℝ^{3} ∣ 2 x - y + 3 z = - 1} .

Determine the intersecting line $E \cap F$ .

Exercise (3 marks)

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

(1, 0, 2), (4, - 3, 2), and (2, 1, - 1)

lie.

Exercise (4 marks)

We consider the linear system

\begin{matrix} 2 x & - a y & = & - 2 \\ a x & + 3 z & = & 3 \\ - \frac{1}{3} x & + y & + z & = & 2 \end{matrix}

over the real numbers, depending on the parameter $a \in ℝ$ . 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

a x + b y = 0

c x + d y = 0

has only the trivial solution $(0, 0)$ if and only if $a d - b c \neq 0$ .

Exercise (4 (2+2) marks)

A system of linear inequalities is given by

x \geq 0,

y + x \geq 0,

- 1 - y \leq - x,

5 y - 2 x \geq 3 .

a) Sketch the solution set of this system.

b) Determine the corners of this solution set.

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