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# 2 equations

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

### 1 Answer by Expert Tutors

Vivian L. | Microsoft Word/Excel/Outlook, essay composition, math; I LOVE TO TEACHMicrosoft Word/Excel/Outlook, essay comp...
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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)