Beginning Algebra With Applications, Chapter 4, 4.1, Section 4.1, Problem 22

The sum of two numbers is fifteen. One less than three times the smaller is equal to the larger. Find the two numbers.

If we let $x$ and $y$ be the smaller and larger numbers, respectively, then we have

$x+y = 15 \qquad$ Equation 1

And

$3x-1 = y \qquad$ Equation 2

By substituting equation 2 with 1, we get


$
\begin{equation}
\begin{aligned}


x + (3x-1) =& 15
\\
\\
x + 3x - 1 =& 15
\\
\\
4x =& 16
\\
\\
\frac{4x}{4} =& \frac{16}{4}
\\
\\
x =& 4

\end{aligned}
\end{equation}
$


Then, by using back substitution, we get


$
\begin{equation}
\begin{aligned}

3(4) -1 =& y
\\
12-1 =& y
\\
11 =& y

\end{aligned}
\end{equation}
$


Therefore, the two numbers are 4 and 11.