Question 1121649
{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2.3,2,"c=?"),
locate(-3.5,-2,"j=?"),
locate(0,-2.7,T),
locate(-.45,-1,"i=?"),
locate(.6,.4,"h=?"), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B),
locate(-1.5,.5,"f=?"),
locate(-.4,2.3,"d=?"),
locate(1.8,2,"e=?"),
locate(-.4,1.1,"g=?") )}}}

In a class of 40 students, 
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That means:

(1) c+d+e+f+g+h+i+j = 40
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17 have ridden an airplane, 
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(2) c+d+f+g = 17
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28 have ridden a boat, 
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(3) d+e+g+h = 28
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10 have ridden a train, 
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f+g+h+i = 10
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12 have ridden both an airplane and a boat. 
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(4) d+g = 12
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3 have ridden a train only 
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(5) i = 3
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and 4 have ridden an airplane only. 
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(6) c = 4
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Some students have not yet ridden any of 
the three modes of transportation
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(7) j > 0 (that is, j is not 0)
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and an equal number have taken all the three. 
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(8) g = j  but we'll need the letters on the left side
so we'll write it as:
(8) g-j = 0
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How many students have used all the three 
modes of transportation? 
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This asks for the value of g.
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How many students have taken only the boat?
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This asks for the value of e.

So we put all equations down and line up the unknowns
[we don't need (7)]

(1) c+d+e+f+g+h+i+j = 40
(2) c+d  +f+g       = 17
(3)   d+e  +g+h     = 28
(4)   d    +g       = 12
(5)             i   =  3
(6) c               =  4
(8)         g       =  j

Change the bottom equation to g-j = 0

(1) c+d+e+f+g+h+i+j = 40
(2) c+d  +f+g       = 17
(3)   d+e  +g+h     = 28
(4)   d    +g       = 12
(5)             i   =  3
(6) c               =  4
(8)         g    -j =  0

We have the 7 x 9 augmented matrix:

{{{(matrix(7,10,

1,1,1,1,1,1,1,1,"|",40,
1,1,0,1,1,0,0,0,"|",17,
0,1,1,0,1,1,0,0,"|",28,
0,1,0,0,1,0,0,0,"|",12,
0,0,0,0,0,0,1,0,"|",3,
1,0,0,0,0,0,0,0,"|",4,
0,0,0,0,1,0,0,-1,"|",0))}}}

Using row operations to reduce this matrix to
row reduced echelon form, we have:

{{{(matrix(7,10,

1,0,0,0,0,0,0,0,"|",4,
0,1,0,0,0,0,0,0,"|",8,
0,0,1,0,0,0,0,0,"|",16,
0,0,0,1,0,0,0,0,"|",1,
0,0,0,0,1,0,0,0,"|",4,
0,0,0,0,0,0,1,0,"|",3,
0,0,0,0,0,0,0,1,"|",4))}}}

Which describes this system:

c = 4
d = 8
e = 16
f = 1
g = 4
i = 3
j = 4

So we have the answers g = 4 and e = 16.

But let's complete the Venn diagram:

We have everything but h.

But we can find h by substituting in

(3)   d+e  +g+h     = 28
      8+16 +4+h     = 28
           28+h     = 28
              h     = 0

So the complete Venn diagram is:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2.3,2,"c=4"),
locate(-3.5,-2,"j=4"),
locate(0,-2.7,T),
locate(-.45,-1,"i=3"),
locate(.6,.4,"h=0"), 
circle(sqrt(2),sqrt(2),2), 
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B),
locate(-1.5,.5,"f=1"),
locate(-.4,2.3,"d=8"),
locate(1.8,2,"e=16"),
locate(-.4,1.1,"g=4") )}}}

Edwin</pre>