#
OEF linear systems
--- Introduction ---

This module contains actually 20 exercises on systems of linear
equations.

### 3 bottles

We have 3 bottles, each containing a certain amount of water. - If we pour cl of water from bottle A to bottle B, B would have times of water as in A.
- If we pour cl of water from bottle B to bottle C, C would have times of water as in B.
- If we pour cl of water from bottle C to bottle A, A would have the same amount of water as C.

How much water there is in each bottle (in centiliters)?

### Equal distance

Find the coordinates of the point p=(x,y) in the cartesian plane, such that: - The distance between p and q
_{1}=(,) equals that between p and q_{2}=(,). >- The distance between p and r
_{1}=(,) equals that between p and r_{2}=(,).

### Intersection of lines

Consider two lines in the cartesian plan, defined respectively by the equations x y = , x y = .

Determine the point p=(x,y) where the two lines meet.

### Four integers II

We have 4 integers *a,b,c,d* such that: - The average of and is .
- The average of and is .
- The average of and is .

What is the average of and ?

### Four integers III

Find 4 integers *a,b,c,d* such that: - The average of and is .
- The average of and is .
- The average of and is .
- The average of and is .

### Four integers

We have 4 integers *a,b,c,d* such that: - The average of
*a*, *b* and *c* is . - The average of
*b*, *c* and *d* is . - The average of
*c*, *d* and *a* is . - The average of
*d*, *a* and *b* is .

What are these 4 integers?

### Vertices of triangle

We have a triangle ABC in the cartesian plane, such that: - The middle of the side AB is (,).
- The middle of the side BC is (,).
- The middle of the side AC is (,).

What are the coordinates of the 3 vertices A, B, C of the triangle? In order to give your reply, we suppose A=(x_{1},y_{1}), B=(x_{2},y_{2}), C=(x_{3},y_{3}).

### Three integers

We have 3 integers *a,b,c* such that: - The average of
*a* and *b* is . - The average of
*b* and *c* is . - The average of
*c* and *a* is .

What are these 3 integers?

### Alloy 3 metals

A factory produces alloy from 3 types of recovered metals. The compositions of the 3 recovered metals are as follows. type | iron | nickel | copper |

metal A | % | % | % |

metal B | % | % | % |

metal C | % | % | % |

The factory has received an order of tons of an alloy with % of iron, % of nickel and % of copper. How many tons of each type of recovered metal should be taken in order to satisfy this order?

### Almost diagonal

Determine the value of _{1} is the solution of the following linear system with equations and variables, for >3. _{1} _{2} | = |

_{2} _{3} | = |

**. . .** |

_{-1} _{} | = |

_{} | = |

(The solution is a function of , which depends on the parity of .)

### Center of circle

Find the center _{0} = (x_{0},y_{0}) of the circle passing through the three points1=(,) , _{2}=(,) , _{3}=(,) .

### Equation of circle

Any circle in the cartesian plane can be described by an equation of the form2+^{2} = ++,

where ,, are real numbers. Find the equation of the circle *C* passing through the 3 points

_{1}=(,) , _{2}=(,) , _{3}=(,) ,

by giving the values for ,,.

### Homogeneous 2x3

Find a non-zero solution of the following homogeneous linear system. The values of x,y,z in your solution should be integers.

### Homogeneous 3x4

Find a non-zero solution of the following homogeneous linear system. The values of x,y,z,t in your solution should be integers.

### Quadrilateral

We have a quadrilateral in the cartesian plane, with 4 vertices ,,,, such that: - The middle of the side is ( , ).
- The middle of the side is ( , ).
- The middle of the side is ( , ).

What is the middle (x,y) of the side ?

### Six integers

We have 6 integers ,,,,, such that: - The average of and is .
- The average of and is .
- The average of and is .
- The average of and is .
- The average of and is .

What is the average of and ?

### Solve 2x2

Find the solution of the following system.

### Solve 3x3

Find the solution of the following system.

### Triangular system

Determine the value of _{1} in the solution of the following linear system with equations and variables, for >3. _{1}+_{2}+_{3}+...+_{} | = |

_{2}+_{3}+...+_{} | = |

**. . .** |

_{-1}+_{} | = |

_{} | = |

### Type of solutions

We have a system of linear in . Among the following propositions, which are true? - A. The system may have no solution.
- B. The system may have a unique solution.
- C. The system may have infinitely many solutions.

Other exercises on:
linear systems
linear algebra

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