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. 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. >
  2. 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: What is the average of and  ?

Four integers III

Find 4 integers a,b,c,d such that:

Four integers

We have 4 integers a,b,c,d such that: What are these 4 integers?

Vertices of triangle

We have a triangle ABC in the cartesian plane, such that: What are the coordinates of the 3 vertices A, B, C of the triangle?

In order to give your reply, we suppose A=(x1,y1), B=(x2,y2), C=(x3,y3).

Three integers

We have 3 integers a,b,c such that: 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.

Quadrilateral

We have a quadrilateral in the cartesian plane, with 4 vertices ,,,, such that: What is the middle (x,y) of the side ?

Six integers

We have 6 integers ,,,,, such that: 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?
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