# How do you solve the following system?: 4x + 5y = 2 , 17 x + 2y = 6

Apr 5, 2017

$x = \frac{26}{77}$
$y = \frac{10}{77}$

#### Explanation:

There are a couple of ways we can solve this. I'm going to use the elimination method
$4 x + 5 y = 2$
$17 x + 2 y = 6$

I need to make two of the variables equal, so I'm going to multiply the second equation by $2.5$. That will change $2 y$ into $5 y$.

$4 x + 5 y = 2$
$2.5 \left(17 x + 2 y = 6\right)$ or $42.5 x + 5 y = 15$

Now we subtract the two equations:
$4 x + 5 y = 2$
$-$
$42.5 + 5 y = 15$
$\textcolor{b l a c k}{- - - - - -}$
$- 38.5 x + 0 y = - 13$

If we simplify our equation, we find that $x = \frac{- 13}{-} 38.5$ or $x = \frac{26}{77}$

Now we just solve for $y$. We can use either of the two equations. I like the first one (it has nicer numbers than $17$).

$4 \left(\textcolor{p u r p \le}{x}\right) + 5 y = 2$
$4 \left(\textcolor{p u r p \le}{\frac{26}{77}}\right) + 5 y = 2$
$\cancel{\frac{104}{77}} + 5 y = 2$
$\cancel{- \frac{104}{77}} \textcolor{w h i t e}{+ 5 y} - \frac{104}{77}$

Now we have
$5 y = \frac{50}{77}$
or
$y = \frac{10}{77}$.

To double check our work, we need to plug our values into one (or both) of the equations.
$4 \left(\frac{26}{77}\right) + 5 \left(\frac{10}{77}\right)$ should equal $2$
$\frac{104}{77} + \frac{50}{77}$
$2 = 2$, so we were right!
Just to be safe, let's look at the other equation.

$17 \left(\frac{26}{77}\right) + 2 \left(\frac{10}{77}\right)$ should equal $6$
$\frac{442}{77} + \frac{20}{77}$
$6 = 6$
Good job, we got it right! Nice work

Apr 5, 2017

Using the process of elimination, we find that $x = 0.338$ and $y = 0.130$. (See explanation)

#### Explanation:

We can solve this system of equations using the elimination method. First, we can set up the system in such a way that the $y$ terms cancel out. To do this, we need both the $y$ terms in each equation to have the same coefficient (one negative and one positive):

$2 \left(4 x + 5 y = 2\right)$
$- 5 \left(17 x + 2 y = 6\right)$

So we get:
$8 x + 10 y = 4$
$- 85 x - 10 y = - 30$

Adding these two equations together gives us:
$- 77 x = - 26$

$x = \frac{26}{77} = 0.338$

Plug this value of $x$ back into one of the given equations:
$4 \left(0.338\right) + 5 y = 2$

$y = \frac{2 - 4 \left(0.338\right)}{5} = 0.130$

So, $x = 0.338$ and $y = 0.130$