# How do you solve the simultaneous equations m+n=19 and m-n=7 ?

Mar 23, 2017

$13$ and $6$

#### Explanation:

Suppose the numbers are $m$ and $n$.

We are given:

$\left\{\begin{matrix}m + n = 19 \\ m - n = 7\end{matrix}\right.$

Adding these two equations together we get:

$2 m = 26$

Dividing both sides by $2$, we find:

$m = 13$

Then from the first equation, we find:

$n = 19 - m = 6$

Mar 23, 2017

See the entire solution process below:

#### Explanation:

First, let's define the two numbers we are looking for. I will call them:

$n$ and $m$.

Next, from the problem we know we can write:

$n + m = 19$ and $n - m = 7$

Then, solve the second equation for $n$;

$n - m = 7$

$n - m + \textcolor{red}{m} = 7 + \textcolor{red}{m}$

$n - 0 = 7 + m$

$n = 7 + m$

Now, substitute $7 + m$ for $n$ in the first equation and solve for $m$:

$n + m = 19$ becomes:

$7 + m + m = 19$

$7 + 2 m = 19$

$- \textcolor{red}{7} + 7 + 2 m = - \textcolor{red}{7} + 19$

$0 + 2 m = 12$

$2 m = 12$

$\frac{2 m}{\textcolor{red}{2}} = \frac{12}{\textcolor{red}{2}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} m}{\cancel{\textcolor{red}{2}}} = 6$

$m = 6$

We have found the first number. To find the second number substitute $6$ for $m$ back into the equation we solved for $m$ and calculate $n$:

$n = 7 + m$ becomes:

$n = 7 + 6$

$n = 13$

The two numbers are $6$ and $13$

$6 + 13 = 19$ and $13 - 6 = 7$