# 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:
1. The distance between p and q1=(,) equals that between p and q2=(,).
2. >
3. The distance between p and r1=(,) equals that between p and r2=(,).

### 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?

### 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.
typeironnickelcopper
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 = (x0,y0) of the circle passing through the three points
1=(,) , 2=(,) , 3=(,) .

### Equation of circle

Any circle in the cartesian plane can be described by an equation of the form
2+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.
 = 0 (1) = 0 (2)
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.
 = 0 (1) = 0 (2) = 0 (3)
The values of x,y,z,t in your solution should be integers.

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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