# Find all solutions to the non-linear systems: 4sqrtx + sqrty = 34 and sqrtx + 4sqrty = 16 ?

Dec 5, 2017

x=64
y=4

#### Explanation:

if
$\sqrt{x} = a$
$\sqrt{y} = b$
then
$4 a + b = 34$
$a + 4 b = 16$
multyply the first equation by -1; the second by 4 and add them together. Then you get

$15 b = 64 - 34$
$b = \frac{30}{15} = 2$
Now. Let's go back to find a.
4a+b=34 so after substitating b=2 you get:
$a = \frac{32}{4} = 8$

then back:
$\sqrt{x} = a$
$\sqrt{x} = 8$
$x = {8}^{2} = 64$

$\sqrt{y} = b$
$\sqrt{y} = 2$
$y = {2}^{2} = 4$

the answers are $\therefore x = 64 \mathmr{and} y = 4$

#### Explanation:

the equations are $4 \sqrt{x} + \sqrt{y} = 34 \to \left(1\right)$
$\sqrt{x} + 4 \sqrt{y} = 16 \to \left(2\right)$
multiplying $\left(2\right) b y - 4$ we get
$- 4 \sqrt{x} - 16 \sqrt{y} = - 64 \to \left(3\right)$
solving $\left(2\right) + \left(3\right)$
$4 \sqrt{x} + \sqrt{y} - 4 \sqrt{x} - 16 \sqrt{y} = - 64 + 34 \Rightarrow - 15 \sqrt{y} = - 30 \Rightarrow \sqrt{y} = 2 \Rightarrow y = 4$
put $y = 4$ in $\left(1\right)$ we get $4 \sqrt{x} + 2 = 34 \Rightarrow 4 \sqrt{x} = 32 \Rightarrow \sqrt{x} = 8 \Rightarrow x = 64$