Suppose a, b, c are nonnegative numbers, and 3a + 2b + c = 5, 2a + b − 3c = 1. Find the Maximum value of S = 3a + b − 7c.

Do you know linear programming?

draw 3 axis graph (x,y,z) is (a, b , c)
Find all the corners of (a,b,c)
when a = 0
(0,0,5) (0,5/2,0)
and
(0 , 0 , -1/3) (0 1, 0)

when b = 0
(0,0,5) ( 5/3,0,0)
and
(1/2,0,0) (0,0,-1/3) note negative

when c = 0
(0,5/2,0) ( 5/3,0,0)
and
(0,1,0) (1/2,0,0)

You have three likely corners
p (5/3, 0, 0)
q (0, 5/2, 0)
r (0, 0, 5 )
try
p
S = 5
try q
S = 5/2
try r
S = negative
so the biggest I got was 5

To find the maximum value of S = 3a + b - 7c, we can use the method of substitution. We'll solve the given system of equations to express a, b, and c in terms of a single variable, and then substitute those expressions into the equation for S.

1. Solve the given system of equations:
3a + 2b + c = 5 ---(1)
2a + b - 3c = 1 ---(2)

To eliminate a variable, we'll multiply equation (2) by 3 and equation (1) by -1:

-3a - 2b - c = -5 ---(3)
6a + 3b - 9c = 3 ---(4)

Now, let's add equations (2) and (4) to eliminate b:
(2a + b - 3c) + (6a + 3b - 9c) = 1 + 3
8a + 4c = 4
Simplify:
2a + c = 1 ---(5)

2. Now we have two equations:
2a + c = 1 ---(5)
3a + 2b + c = 5 ---(1)

To express a and b in terms of c, we'll solve these equations simultaneously. Subtract equation (5) from equation (1):
(3a + 2b + c) - (2a + c) = 5 - 1
a + 2b = 4

Now, we have two equations:
2a + c = 1 ---(5)
a + 2b = 4

Solve equation (5) for a:
2a = 1 - c
a = (1 - c)/2

Substitute the value of a into equation (6):
[(1 - c)/2] + 2b = 4
Simplify:
1 - c + 4b = 8
4b = 7 + c
b = (7 + c)/4

3. We now have expressions for a and b in terms of c. Substitute these expressions into the equation for S:

S = 3a + b - 7c
= 3[(1 - c)/2] + [(7 + c)/4] - 7c
= (3 - 3c)/2 + (7 + c)/4 - 7c
= (6 - 6c)/4 + (7 + c)/4 - (28c/4)
= (6 - 6c + 7 + c - 28c)/4
= (-23c + 13)/4

4. To maximize S, we need to find the maximum value of c that satisfies the given conditions. Since a, b, and c are nonnegative numbers, we need to ensure that c is also nonnegative.

To determine the maximum value of c, consider the range of values that satisfy equation (5):
2a + c = 1
c = 1 - 2a

Since a and c are nonnegative, the maximum value of a is when a = 0. Substituting a = 0 into equation (7), we can find the maximum value of c:
c = 1 - 2(0)
c = 1

Therefore, the maximum value of c that satisfies the given conditions is c = 1.

5. Finally, substitute c = 1 into the expression for S to find the maximum value:
S = (-23(1) + 13)/4
= (-23 + 13)/4
= -10/4
= -2.5

So, the maximum value of S = 3a + b - 7c is -2.5.