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

*Exercises*

and are the members of one family. In this case, is three times as old as and together, is older than , and is older than , moreover, the age difference between and is twice as large as the difference between and . Furthermore, is seven times as old as , and the sum of the ages of all family members is equal to the paternal grandmother's age, that is .

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

b) Solve this system of equations.

Kevin pays € for a winter bunch of flowers with snowdrops and mistletoes, and Jennifer pays € for a bunch with snowdrops and mistletoes. How much does a bunch with one snowdrop and mistletoes cost?

Show that the system of linear equations

has only the trivial solution .

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?

Solve the linear equation

Solve the system of linear equations

Solve the following system of inhomogeneous linear equations.

Does there exist a solution for the system of linear equations

from Example 21.1 ?

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

Bring the system of linear equations

into a standard form, and solve it.

Exhibit a linear equation for the straight line in , which runs through the two points and .

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

defined on a subset
and two distinct points
,
the line through
and
is called the
*secant*
of at

Determine an equation for the secant of the function

to the points and .

Determine a linear equation for the plane in , where the three points

lie.

Given a complex number

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

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

Let be the field with two elements. Solve in the inhomogeneous linear system

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.

Solve the system of linear equations

Solve the system of linear equations

Solve the system of linear equations

Solve the system of linear equations

Determine, in dependence of the parameter , the solution space of the system of linear equations

A system of linear inequalities is given by

Sketch the solution set of this system of inequalities.

Let

be a system of linear inequalities, whose solution set is a triangle. How does the solution set look like, when we replace one inequality by ?

*Hand-in-exercises*

### Exercise (4 marks)

Solve the following system of inhomogeneous linear equations.

### Exercise (3 marks)

Solve the system of linear equations in the variables , which is given by the two equations

and

### Exercise (3 marks)

Consider in the two planes

Determine the intersecting line .

### Exercise (3 marks)

Determine a linear equation for the plane in , where the three points

lie.

### Exercise (4 marks)

We consider the linear system

over the real numbers, depending on the parameter . For which does the system of equations have no solution, one solution or infinitely many solutions?

### Exercise (4 marks)

Show that a system of linear equations

has only the trivial solution if and only if .

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

A system of linear inequalities is given by

a) Sketch the solution set of this system.

b) Determine the corners for this solution set.

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