# Simoultaneous equations

#### darksupernova

Hey there,

Getting very stuck on solving these, ive tried using substitution but i dont know how to rearrange it...
can you help?

$$\displaystyle 137.9 = x + \frac{y}{\sqrt{0.05}}$$

and

$$\displaystyle 275.8 = x + \frac{y}{\sqrt{0.007}}$$

#### Bacterius

Hi,
to rearrange for $$\displaystyle y$$ in the first equation, you can substract $$\displaystyle x$$ then multiply by $$\displaystyle \sqrt{0.05}$$. So you find :

$$\displaystyle 275.8 = x + \frac{\sqrt{0.05}(137.9 - x)}{\sqrt{0.007}}$$

So :

$$\displaystyle 275.8 = x + \frac{\sqrt{0.05}}{\sqrt{0.007}} (137.9 - x)$$

Setting $$\displaystyle a = \frac{\sqrt{0.05}}{\sqrt{0.007}}$$ to make it a bit simpler, we are left with :

$$\displaystyle 275.8 = x + a (137.9 - x)$$

So :

$$\displaystyle 275.8 = x + 137.9a - ax$$

$$\displaystyle 275.8 - 137.9a = x - ax$$

$$\displaystyle 275.8 - 137.9a = x(1 - a)$$

$$\displaystyle \frac{275.8 - 137.9a}{1 - a} = x$$

Substituting back the value we chose for $$\displaystyle a$$, we get :

$$\displaystyle x = \frac{275.8 - 137.9 \left ( \frac{\sqrt{0.05}}{\sqrt{0.007}} \right )}{1 - \frac{\sqrt{0.05}}{\sqrt{0.007}}} \approx 55.454$$

Finding the value of $$\displaystyle y$$ is now straightforward Does it make sense ? Remember not to get stopped by impressive square roots and stuff : as long as there are no $$\displaystyle x$$ terms in them, they can be considered a constant (and thus substituted to some letter, $$\displaystyle a$$ in my example)

#### darksupernova

ah yes that helps thanks very much!
i see the way you got the unkown terms on one side, very nice Thanks again,

Max

• Bacterius

#### Wilmer

Hey there,

Getting very stuck on solving these, ive tried using substitution but i dont know how to rearrange it...
can you help?

$$\displaystyle 137.9 = x + \frac{y}{\sqrt{0.05}}$$

and

$$\displaystyle 275.8 = x + \frac{y}{\sqrt{0.007}}$$
Since 275.8 is double 137.9, then you can go this simpler way:

x + y/sqrt(.007) = 2x + 2y/sqrt(.05)

#### HallsofIvy

MHF Helper
Seeing "x" alone in both equations, the first thing I would think of is subtracting one equation from the other:
$$\displaystyle 275.8 = x + \frac{y}{\sqrt{0.007}}$$
$$\displaystyle 137.9 = x + \frac{y}{\sqrt{0.05}}$$

$$\displaystyle 137.9= \frac{y}{\sqrt{0.007}}- \frac{y}{\sqrt{0.05}}= y\left(\frac{1}{\sqrt{0.007}}- \frac{1}{\sqrt{0.05}}\right)$$

$$\displaystyle 137.9= y\left(\frac{\sqrt{0.05}- \sqrt{0.007}}{\sqrt{(0.05)(.007)}}\right)$$

$$\displaystyle y= \frac{137.9\sqrt{0.00035}}{\sqrt{0.05}- \sqrt{0.007}}$$