2 equations

Question

A student has some $1 bills and some $5 bills totaling $47. What 2 equations using substitutions can be used to solve?

Answer 1

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)