SOLUTION: The total expenditure E of a household is partly constant and partly varies directly as the size S of the household. (a) find the equation connecting E and S, if when E=300,S=5per

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Question 1196618: The total expenditure E of a household is partly constant and partly varies directly as the size S of the household.
(a) find the equation connecting E and S, if when E=300,S=5persons and E=400,S=6
(b) what is the total expenditure when the size of the household is 10persoms
(c) how many persons will consume a total expenditure of $600
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
if the total expenditure E of a household is partly constant and partly varies directly as the size S of the household, we have

E+=+a+%2B+bS+ were a is a constant and b the number of persons

a) find the equation connecting E and S, if when

E=300,S=5 persons

300+=+a+%2B+5b.................solve for+a
a=300-5b...........eq.1


and+E=400,S=6
400+=+a+%2B+6b.................solve for a
a=400-6b...........eq.2

then from eq.1 and eq.2 we have

300-5b=400-6b..............solve for b
6b-5b=400-300
b=100

go to
a=400-6b...........eq.2, substitute+b
a=400-6%2A100
a=400-600
a=+-200

and equation is
E+=-200%2B+100S


b.
if S=10
E+=-200%2B+100%2A10=800
the total expenditure is $800 when the size of the household is 10 persons

c.
E+= $600
600=-200%2B+100S
600%2B200=100S........divide by 100
6%2B2=S
S=8
8 persons will consume a total expenditure of $600

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

The total expenditure E of a household is partly constant and partly varies directly as the size S of the household.
(a) find the equation connecting E and S, if when E=300,S=5persons and E=400,S=6
(b) what is the total expenditure when the size of the household is 10persoms
(c) how many persons will consume a total expenditure of $600
This is a LINEAR system.

Let slope be m, and constant, b
We then get: E = mS + b____300 = 5m + b ----- eq (i)
             E = mS + b____400 = 6m + b ----- eq (ii) 
                           100 = m ----- Subtracting eq (i) from eq (ii)
                           300 = 5(100) + b ---- Substituting 100 for m in eq (i)
                     300 - 500 = b____- 200 = b

   Linear equation: E = mS + b______E = 100S - 200
                                    E = 100(10) - 200 --- Substituting 10 for S
For 10 people, expenditure or E = 1,000 - 200 = $800

   Linear equation: E = mS + b______E = 100S - 200
                                  600 = 100S - 200 ------ Substituting 600 for E
                            600 + 200 = 100S
                                  800 = 100S
Number of persons when expenditure is $600 or