# What are two numbers with a sum of 35 and a difference of 7?

Jul 15, 2016

Make a system of equations using the given information and solve to find the numbers are $21$ and $14$.

#### Explanation:

The first thing to do in algebraic equations is to assign variables to what you don't know. In this case, we don't know either number so we'll call them $x$ and $y$.

The problem gives us two key bits of info. One, these numbers have a difference of $7$; so when you subtract them, you get $7$:
$x - y = 7$

Also, they have a sum of $35$; so when you add them, you get $35$:
$x + y = 35$

We now have a system of two equations with two unknowns:
$x - y = 7$
$x + y = 35$

If we add them together, we see we can cancel the $y$s:
$\textcolor{w h i t e}{X} x - y = 7$
$+ \underline{x + y = 35}$
$\textcolor{w h i t e}{X} 2 x + 0 y = 42$
$\to 2 x = 42$

Now divide by $2$ and we have $x = 21$. From the equation $x + y = 35$, we can see that $y = 35 - x$. Using this and the fact that $x = 21$, we can solve for $y$:
$y = 35 - x$
$\to y = 35 - 21 = 14$

So the two numbers are $21$ and $14$, which do indeed add to $35$ and have a difference of $7$.