System of equations

absvalue · #1 $Old$ November 16th, 2009, 08:33 PM

Could someone point me in the right direction? I've been wrestling with this one for a while.

$100x + 200y + 500z = 47$
$350x + 5y + 250z = 33.9$
$200x + 80y + 100z = 23.4$

11rdc11 · #2 $Old$ November 16th, 2009, 08:59 PM

Quote:

Originally Posted by absvalue $View Post$

Could someone point me in the right direction? I've been wrestling with this one for a while.

$100x + 200y + 500z = 47$
$350x + 5y + 250z = 33.9$
$200x + 80y + 100z = 23.4$

multiply the 2nd equation by -2 and get

$-700x -10y -500z = -67.80$

Now cancel out the z in the 1st and new equation we got

$-600x +190y = -20.80$

Now multiply equation 2 by 2 and multiply equation 3 by -5 to create 2 new equations.

Now add the two new equations and the z's should cancel out leaving

$-300x -390y = -49.20$

Can you finish up from here?

Soroban · #3 $Old$ November 16th, 2009, 09:14 PM

Hello, absvalue!

It can be done with basic algebra IF we're careful . . .

Quote:

$\begin{array}{cccc}100x + 200y + 500z &=& 47 & [1] \\ 350x + 5y + 250z &=& 33.9& [2] \\ 200x + 80y + 100z &=& 23.4 & [3] \end{array}$

$\begin{array}{cccccc}\text{Multiply }2 \times [2]\!: & 700x + 10y + 500z &=&67.8 \\ \text{Subtract [1]:} & \text{-}100x - 200y - 500z &=& \text{-}47 \end{array}$

We have: . $600x - 190y \:=\:20.8 \;\;[4]$

$\begin{array}{cccccc} \text{Multiply }5 \times [3]\!: &1000x + 400y + 500z &=& 117 \\ \text{Subtract [1]:} & \text{-}100x - 200y - 500z &=& \text{-}47 \end{array}$

We have: . $900x + 200y \:=\:70\;\;[5]$

$\begin{array}{ccccc} \text{Multiply }2 \times [5]\!: & 1800x + 400y &=& 140 \\ \text{Multiply }3\times[4]\!: & 1800x - 570y &=& 62.4 \end{array}$

Subtract: . $970y \:=\:77.6 \quad\Rightarrow\quad \boxed{y \:=\:0.08}$

Substitute into [5]: . $900x + 200(0.08) \:=\:70 \quad\Rightarrow\quad \boxed{x \:=\:0.06}$

Substitute into [3]: . $200(0.06) + 80(0.08) + 100z \:=\:23.4 \quad\Rightarrow\quad \boxed{z \:=\:0.05}$

absvalue · #4 $Old$ November 17th, 2009, 08:04 AM

Thanks, that makes sense now.

You guys are awesome.