Answer
The number of times the word “love” is used is 630, number of times the word “thanks” is used is 243, number of times the word “sorry” is used is 211.
Work Step by Step
Let the number of times the word “love” is used be l, number of times the word “thanks” is used is t, number of times the word “sorry” is used is s, and the total number of times the word “love”, “thanks”, and “sorry” are used be p.
The usage of the word “love” was used exceeded the usage of “sorry” by 419, this is stated algebraically as,
\[l-s=419\] (I)
The usage of “thanks” exceeded the usage of “sorry” by 32, this is stated algebraically as,
\[t-s=32\] (II)
Multiplying (II) with\[-1\],
(III)
The total number of times the word “love”, “thanks”, and “sorry” are used is 1084, this is stated algebraically as,
\[\begin{align}
& p=l+s+t \\
& l+s+t=1084 \\
& p=1084
\end{align}\] (IV)
Solve (I) and (II) to get,
\[\frac{\begin{align}
& l-s=419 \\
& -t+s=-32
\end{align}}{l-t=387}\]
\[\ l-t=387\] (V)
Solve (IV) and (V) to get,
\[\frac{\begin{align}
& l+s+t=1048 \\
& l-t=387
\end{align}}{2l+s=1471}\]
\[\ \ 2l+s=1471\] (VI)
Solve (VI) and (I) to get,
\[\frac{\begin{align}
& \ \ 2l+s=1471 \\
& l-s=419 \\
\end{align}}{3l=1890}\]
\[\ \ \ 3l=1890\] (VII)
Solving the expression in (VII), to get,
\[\begin{align}
& 3l=1890 \\
& l=\frac{1890}{3} \\
& l=630
\end{align}\]
The number of times the word “love” is used is 630. Put this value of l in (I) to get,
\[\begin{align}
& l-s=419 \\
& 630-s=419 \\
& s=630-419 \\
& s=211
\end{align}\]
The number of times the word “sorry” is used is 211. Put this value of s in (II) to get,
\[\begin{align}
& t-s=32 \\
& t-211=32 \\
& t=211+32 \\
& t=243
\end{align}\]
The number of times the word “thanks” is used is 243.
The number of times the word “love” is used is 630, number of times the word “thanks” is used is 243, number of times the word “sorry” is used is 211.
