Kevin wishes to buy apples and bananas, Apples are 50 cents per pound and bananas are 10 cents per pound. Kevin will spend $5.00 for his fruit. How do you write an equation that models this situation and describe the meaning of the two intercepts?

#color(blue)("Initial build of equation")#

Let count of apples be: #" "a#
Let count of bananas be:#" " b#

Cost of apples per pound (lb) is: #" "$0.50#
Cost of bananas per pound (lb) is: #" "$0.10#

Let total cost be:#" " t#

Then #" "t=$0.5a+$0.1b#

Given that total cost #(t)# is $5.00 we have:

#t=$0.5a+$0.1b" "->" "$5.00=$0.5a+$0.1b#

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#color(blue)("Build the model")#

The counts of apples or bananas are not specified so within limits of total cost we can only have so many of each of them. The proportion is controlled by the total cost of $5

#color(red)("Assumption: we are required to model quantities")#

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If all apples then the maximum count is for $5 worth:

#=>a=($5.00)/($0.5) = 10# as a maximum

Thus #b# would have the count of #b=0# for this condition
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If all bananas then the maximum count is for $5 worth:

#=>b=($5.00)/($0.1) = 50# as a maximum

Thus #a# would have the count of #a=0# for this condition

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#color(brown)("The count of one of them infers the count of the other through the limitation of cost")#

Using this limiting factor we have: #color(brown)(" "$5.00=$0.5a+$0.1b)#

As we are just dealing with counts drop the $ sign

Subtract 0.1b from both sides

#0.5a=5-0.1b#

Lets get rid of the decimals: multiply both sides by 10

#5a=50-b#

Divide both sides by 5

#a=50/5-b/5#

#" "color(blue)(bar(ul(|" Model "->a=10-b/5" "|))#

#color(red)("x- intercept is the condition of all bananas and no apples")#
#color(red)("y- intercept is the condition of all apples and no bananas")#