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2 equations

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Hi Joe;
JOE, PLEASE CONFIRM THAT YOU UNDERSTAND THIS.  I MADE ONE UPDATE BECAUSE I WAS NOT COMPLETELY SATISFIED WITH MY ANSWER.
FIRST EQUATION...
x=quantity of $1 bills
y=quantity of $5 bills
 
FIRST EQUATION...
$47=$1x+$5y
To simplify, let's divide both sides by the symbol $.
$47/$=($1x+$5y)/$
47=x+5y
 
SECOND EQUATION...
Because y is the larger number, we must understand it as it uniquely relates to the number 47...
47 ? 5y
47/5 ? y
9.4 ? y
Obviously, there is no such as thing as 4/10th of a $5 bill.  Therefore
SECOND EQUATION...
y ≤ 9
This student has 9 or fewer $5 bills.
 
LET'S SUBSTITUTE...
47=x+5y
47=x+(5(≤9))
47=x+(≤45)
 
We must proceed to isolate x by subtracting (≤45) from both sides...
47-(≤45)=x+(≤45)-(≤45)
47-(≤45)=x
When we subtract ≤, we must reverse it to ≥...
≥2=x
x≥2
 
This student has at least 2 $1 bills.
 
This student has at most 9 $5 bills.
 
Let's check our math...
$47=$1x+$5y
$47=$1(≥2)+$5(≤9)