Problem 15.7.6

Plot[{-2x+4,-2x+7,x-2,x+1},{x,-4,4}]

x=(u+v)/3;
y=(v-2u)/3;
Simplify[y==x+1]
Simplify[y==x-2]
Simplify[y==-2x+4]
Simplify[y==-2x+7]

Simplify[2(x^2)-x*y-(y^2)]

Integrate[Integrate[u*v/3,{u,-1,2}],{v,4,7}]

49-16

Problem 15.7.7

Plot[{-3/2x+1,-3/2x+3,-1/4 x,-1/4 x +1},{x,-4,4}]

x=(2u-v)/5;
y=(3v-u)/10;
Simplify[y==-3/2x+1]
Simplify[y==-3/2x+3]
Simplify[y==-1/4 x]
Simplify[y==-1/4 x +1]

ContourPlot[{u==2,u==6,v==0,v==4},{u,-10,10},{v,-10,10}]

Simplify[3x^2+14x y+8y^2]

Integrate[Integrate[u*v/10,{u,2,6}],{v,0,4}]

Problem 15.7.10

ContourPlot[{x==1,x==2,y==1,y==2},{x,0,4},{y,0,4}]

x=u;
y=u*v;
Simplify[x==1]
Simplify[x==2]
Simplify[y==1]
Simplify[y==2]

ContourPlot[{u==1,u==2,u*v==1,u*v==2},{u,0,4},{v,0,4}]

Simplify[y/x]

Clear[x,y]
Integrate[Integrate[y/x,{x,1,2}],{y,1,2}]

Clear[x,y]
Integrate[Integrate[u*v,{v,1/u,2/u}],{u,1,2}]