source("http://macosa.dima.unige.it/r.R") # If I have not already loaded the library ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------S05Equation systemseqSystem(x) solves the linear system xsolut[n] is the n-th solution (if before "eqSystem" I putnoPrint=1, the solutions are # not printed and can only be seen with thesolutcommand).## (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 = 4.375S = c( 80,70,60, 74, 17.5,25,30, 21.625, 2.5, 5,10, 4.375 )eqSystem(S)#0.55 0.30 0.15solut[2] # 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".# Theintersection of the three planesx-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 -6.374150fraction(solut[1:4]) # -220/147 96/49 198/49 -937/147 # # Fornonlinear 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 leftOther examples of use