# The sum of two numbers is 51. Their difference is 13. What is the smaller of the two numbers?

Mar 4, 2018

The smaller number is $19$.

#### Explanation:

Translate the two statements from English to math.

For example, "sum" means adding two numbers, "difference" means subtracting two numbers, and "is" means equals. This is how it looks:

$\stackrel{x + y}{\overbrace{\text{The sum of two numbers"" " stackrel= overbrace"is"" " stackrel(51;) overbrace"51."" " stackrel(x-y) overbrace"Their difference"" " stackrel= overbrace"is"" " stackrel13 overbrace"13"".}}}$

Now, make a system of equations:

$\textcolor{w h i t e}{=} \left\{\begin{matrix}x + y = 51 & q \quad q \quad \left(1\right) \\ x - y = 13 & q \quad q \quad \left(2\right)\end{matrix}\right.$

Use equation $\left(2\right)$ to solve for (a temporary) $x$ value:

$\textcolor{w h i t e}{\implies} x - y = 13$

$\textcolor{w h i t e}{\implies} x \textcolor{red}{\cancel{\textcolor{b l a c k}{- y + y}}} = 13 + y$

$\textcolor{w h i t e}{\implies} x = 13 + y$

Next, plug this value of $x$ into equation $\left(1\right)$:

$\textcolor{w h i t e}{\implies} x + y = 51$

$\implies \left(13 + y\right) + y = 51$

$\textcolor{w h i t e}{\implies} 13 + 2 y = 51$

$\textcolor{w h i t e}{\implies} 2 y = 51 - 13$

$\textcolor{w h i t e}{\implies} 2 y = 38$

$\textcolor{w h i t e}{\implies} y = \frac{38}{2}$

$\textcolor{w h i t e}{\implies} y = 19$

Lastly, plug this value of $y$ into either equation (I'll do $\left(1\right)$) and solve for $x$:

$\textcolor{w h i t e}{\implies} x + y = 51$

$\implies x + 19 = 51$

$\textcolor{w h i t e}{\implies} x = 51 - 19$

$\textcolor{w h i t e}{\implies} x = 32$

The two numbers in the word problem are $19$ and $32$. Since the problem is asking for the smaller of the two numbers, the answer is $19$.