```source("http://macosa.dima.unige.it/r.R")    # If I have not already loaded the library
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S 05 Equation systems

eqSystem(x) solves the linear system x
solut[n] is the n-th solution (if before "eqSystem" I put noPrint=1, the solutions are
# not printed and can only be seen with the solut command).

## (I have three metal bars made of silver, copper and tin. Their compositions are:
##    silver copper  tin
## 1^  80%  17.5%   2.5%    How much metal do you need to take from each bar to get
## 2^  70%    25%     5%    a compound metal in the following way?
## 3^  60%    30%    10%      silver: 74%    copper: 21.625%    tin: 4.375%

## Let's solve the system
#   80x + 70y + 60z = 74
# 17.5x + 25y + 30z = 21.625
#  2.5x + 5y +  10z = 4.375

S = c(
80,70,60, 74,
17.5,25,30, 21.625,
2.5, 5,10, 4.375  )
eqSystem(S)
# 0.55 0.30 0.15
solut # To display only the 2nd solution (y)
# 0.3

## 55% must be taken from the first bar, 30% from the second, 15% from the third

# If the system has no solution or has infinitely many solutions it appears "singular
# system".

# The intersection of the three planes x-y-4z=3, x+y-2z=1, 3x+6y-z=0
Q = c(1,-1,-4,3, 1,1,-2,1, 3,6,-1,0); eqSystem(Q)
# 2 -1 0
# The point of intersection is (2,-1,0)

# Another example. The linear system:
#  2x-5y+4z+ u = -3
#   x-2y+ z- u =  5
#   x+   6z+2u = 10
# 12x+4y+3z    =  2
S = c(
2,-5, 4, 1, -3,
1,-2, 1,-1,  5,
1, 0, 6, 2, 10,
12, 4, 3, 0,  2)
eqSystem(S)
# -1.496599  1.959184  4.040816 -6.374150
fraction(solut[1:4])
#  -220/147    96/49    198/49  -937/147
#
# For nonlinear systems, you can go graphically and find solutions up to 6 or 7 digits.
# But when you can relate to an equation in one variable, you can use equation
# resolution. Let's see a simple example:
#   Find the intersections between  2y+3x-15=0 (line) and (x-3)^2+(y-5)^2=4 (circle)
f = function(x) (15-3*x)/2; g = function(x,y) (x-3)^2+(y-5)^2-4
PLANE(0,7, 0,7); CURVE(g, "blue"); graph( f, 0,7, "blue")
# I get the chart below to the left: # To find the intersections I consider:
h = function(x) g(x, f(x)); graphF( h, 0,4, "blue")

#       [ h(x)=0 when f(x)=y and g(x,y)=0 ]

# I get the chart above to the right, where I see the "x" solutions better
solution(h,0, 0,2)
# 1.153846
more(solution(h,0, 0,2))
# 1.15384615384615
fraction(solution(h,0, 0,2))
# 15/13
solution(h,0, 2,4)
# 3
# The x are 15/13 and 3, I find the y:

f(15/13); f(3)
# 5.769231  3
fraction( f(15/13) )
# 75/13

# The solutions are (x,y) = (15/13, 75/13) and (x,y) = (3,3), which I can control on
# the graph above, on the left

Other examples of use

```