# How do you solve 4m+n=2 and m-4n=9 using substitution?

Apr 27, 2016

$m = 1 \text{ ; } n = - 2$

#### Explanation:

Given:
$4 m + n = 2$ -------------------------------Equation (1)
$m - 4 n = 9$..........................................Equation (2)

$\textcolor{b l u e}{\text{Determine the value of n}}$

$\textcolor{b r o w n}{\text{Showing every step in the first part to demonstrate method}}$

$\textcolor{g r e e n}{\text{Consider Equation (2)}}$

$\textcolor{g r e e n}{\text{Add 4n to both sides}}$

$\text{ } m - 4 n + 4 n = 9 + 4 n$

$\textcolor{g r e e n}{\text{But } 4 n - 4 n = 0}$

$\text{ } m = 9 + 4 n$ .....................................(3)

$\textcolor{g r e e n}{\text{Substitute for m in (1) using (3)}}$

$\text{ "4m+n=2" "->" } 4 \left(9 + 4 n\right) + n = 2$

$\textcolor{g r e e n}{\text{Multiply out the bracket}}$
$\text{ } \implies 36 + 16 n + n = 2$

$\text{ } 36 + 17 n = 2$

$\textcolor{g r e e n}{\text{Subtract 36 from both sides}}$
$\text{ } 17 n = - 34$

$\textcolor{g r e e n}{\text{Divide both sides by 17}}$
$\text{ } \textcolor{b l u e}{n = - \frac{34}{17} = - 2}$
'.........................................................
$\textcolor{b l u e}{\text{Determine the value of m}}$

Substitute $n = - 2$ into (2)

$m - 4 n = 9 \text{ "->" } m - 4 \left(- 2\right) = 9$

$m + 8 = 9$

$m = 9 - 8$

$m = 1$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Check
$4 m + n = 2 \text{ "->" } 4 \left(1\right) + \left(- 2\right) = 2$
$m - 4 n = 9 \text{ "->" } 1 - 4 \left(- 2\right) = 9$