How do you solve 6m + n = 21 and m - 8n = 28 using substitution?

Mar 22, 2018

$n = - 3 , \text{ } m = 4$

Explanation:

Take the second equation:

$m - 8 n = 28$

If we add $8 n$ to both sides, we get:

$m = 8 n + 28$

Now we have two equivalent expressions: $\textcolor{red}{m}$ and $\textcolor{b l u e}{8 n + 28}$.

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Now, look back at the first equation:

$6 m + n = 21$

To solve by substitution, we need to replace (or substitute) one equivalent expression for the other so that we can solve this equation. Let's use the expressions from earlier:

$6 \textcolor{red}{m} + n = 21$

Remember that $\textcolor{red}{m} = \textcolor{b l u e}{8 n + 28}$, so we can do this:

$6 \left(\textcolor{b l u e}{8 n + 28}\right) + n = 21$

Now, we can use simple algebra to solve for $n$.

$6 \left(8 n + 28\right) + n = 21$

$6 \cdot 8 n + 6 \cdot 28 + n = 21$

$48 n + 168 + n = 21$

$49 n + 168 = 21$

$49 n = - 147$

$n = - 3$

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Finally, let's plug $n$ back into one of the equations and solve for $m$ to finish the problem. I'm using equation 1 to solve for $m$, but either one will work just fine.

$6 m + n = 21$

Remember from above that $\textcolor{\mathmr{and} a n \ge}{n} = \textcolor{\lim e g r e e n}{- 3}$

$6 m + \textcolor{\mathmr{and} a n \ge}{n} = 21$

$6 m + \textcolor{\lim e g r e e n}{\left(- 3\right)} = 21$

$6 m - 3 = 21$

$6 m = 24$

$m = 4$

So now we have both parts of our solution! Here's the final solution:

$n = - 3 , \text{ } m = 4$