# If m - n = 32 and 3m - 8n - 6 = 0, then what is 4m - 9n?

May 16, 2018

$4 m - 9 n = 38$

#### Explanation:

First, we solve for $m$ in terms of $n$, using equation 1:

$m - n = 32$

color(blue)(m=n+32

Next we substitute our solution for $m$ in equation 2 to solve for $n$:

$3 m - 8 n - 6 = 0$

$3 \textcolor{b l u e}{\left(n + 32\right)} - 8 n - 6 = 0$

$3 n + 96 - 8 n - 6 = 0$

$90 - 5 n = 0$

$90 = 5 n$

$\textcolor{p u r p \le}{n = 18}$

Now, we can solve equation 3 without ever getting the discrete solution for $m$:

$4 \textcolor{b l u e}{m} - 9 \textcolor{p u r p \le}{n} = x$

$4 \textcolor{b l u e}{\left(\textcolor{p u r p \le}{n} + 32\right)} - 9 \textcolor{p u r p \le}{n} = x$

$4 \textcolor{p u r p \le}{n} + 128 - 9 \textcolor{p u r p \le}{n} = x$

$128 - 5 \textcolor{p u r p \le}{n} = x$

$128 - 5 \textcolor{p u r p \le}{\left(18\right)} = x$

$128 - 90 = x$

$\textcolor{g r e e n}{x = 38}$