#1)
f(x)=2*x^2
g(x)=x^4-2*x^2
show(solve(f(x)==g(x),x))
show(g(-2))
show(g(2))
show(g(0))
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var('y')
solve([y==2*x^2,y==x^4-2*x^2],x,y)
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plot([f(x),g(x)],-3,3)
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#2)
equ1=7*x+12*y==63;show(equ1)
equ2=2*x+3*y==15;show(equ2)
equ2=-4*equ2;show(equ2)
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equ3=equ1+equ2;show(equ3)
show(solve(equ3,x))
solve([equ1,x==-3],x,y)
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show(solve(equ1,y)[0].rhs())
show(solve(equ2,y)[0].rhs())
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plot([solve(equ1,y)[0].rhs(),solve(equ2,y)[0].rhs()],-5,5)
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#3) by matrix algebra
a=matrix([(2,-1),(6,8)]);show(a)
b=matrix(2,1,[2,39]);show(b)
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show(a^(-1))
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show(a^(-1)*b)
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#3) by Cramer's Rule
a=matrix([(2,-1),(39,8)]);show(a)
b=matrix([(2,2),(6,39)]);show(b)
c=matrix([(2,-1),(6,8)]);show(c)
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detA=det(a);show(detA)
detB=det(b);show(detB)
detC=det(c);show(detC)
x=detA/detC;show(x)
y=detB/detC;show(y)
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#3) by augmented matrix
a=matrix(QQ,2,2,[(2,-1),(6,8)]);show(a)
b=matrix(QQ,2,1,[2,39]);show(b)
c=a.augment(b,subdivide=True);show(c)
c.rescale_row(0, 1/2);show(c)
c.add_multiple_of_row(1,0,-6);show(c)
c.rescale_row(1, 1/11);show(c)
c.add_multiple_of_row(0,1,1/2);show(c)
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#4)
var('x,y')
equ1=2*x+y==100;show(equ1)
show(solve(equ1,y)[0].rhs())
equ2=x+y==75;show(equ2)
show(solve(equ2,y)[0].rhs())
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plot([solve(equ1,y)[0].rhs(),solve(equ2,y)[0].rhs()],0,100,fill=true,aspect_ratio=1)
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