# x+y+z=-1, 3x+y+4z=8,-x-y+7z=9?

Nov 9, 2017

$x = 3$
$y = - 5$
$z = 1$

#### Explanation:

There are three equations with three variables.

Make $y$ the subject in all three equations:

$y = - x - z - 1 \text{ }$..... equation 1
$y = - 3 x - 4 z + 8 \text{ }$ ... equation 2
$y = - x + 7 z - 9 \text{ }$...equation 3

By equating the equations in pairs we can form two equations with the variables $x \mathmr{and} z$ and solve them simultaneously

Using equations 1 and 2: $\text{ } y = y$

$\text{ } - x - z - 1 = - 3 x - 4 z + 8$
$3 x - x + 4 z - z = 8 + 1 \text{ } \leftarrow$ re-arrange

$2 x + 3 z = 9 \text{ }$ equation A

Using equations 3 and 2 $\text{ } y = y$

$\text{ "-x+7z-9=-3x-4z+8" } \leftarrow$ re-arrange

$3 x - x + 7 z + 4 z = 8 + 9$

$2 x + 11 z = 17 \text{ }$ equation B

Now solve A and B for $x \mathmr{and} z$

$\text{ } 2 x + 11 z = 17 \textcolor{w h i t e}{m m m m m m m m m m m} A$
$\text{ } 2 x + 3 z = 9 \textcolor{w h i t e}{m m m m m m m m m m m m} B$

$A - B : \text{ } 8 z = 8$
$\textcolor{w h i t e}{m m m m m m} z = 1$

$2 x + 3 \left(1\right) = 9$
$2 x + 3 = 9$
$2 x = 6$
$x = 3$

Now find $y$ from equation 1
$y = - x - z - 1$
$y = - \left(3\right) - \left(1\right) - 1$
$y = - 5$

Check with equation 2

$y = - 3 x - 4 z + 8$
$y = - 3 \left(3\right) - 4 \left(1\right) + 8$
$y = - 9 - 4 + 8$
$y = - 5$

Nov 9, 2017

$x = 3$, $y = - 5$ and $z = 1$

#### Explanation:

$x + y + z = - 1$, $3 x + y + 4 z = 8$ an $- x - y + 7 z = 9$

From first equation, $z = - x - y - 1$

Plug $z$ into second an third ones;

$3 x + y + 4 \cdot \left(- x - y - 1\right) = 8$

$3 x + y - 4 x - 4 y - 4 = 8$

$- x - 3 y = 12$

$- x - y + 7 \cdot \left(- x - y - 1\right) = 9$

$- x - y - 7 x - 7 y - 7 = 9$

$- 8 x - 8 y = 16$

$- 8 \cdot \left(x + y\right) = 16$ or $x + y = - 2$

From second one, $x = - 3 y - 12$

Plug $x$ into third one;

$\left(- 3 y - 12\right) + y = - 2$

$- 2 y - 12 = - 2$

$- 2 y = 10$, so $y = - 5$

Hence $x = - 3 y - 12 = \left(- 3\right) \cdot \left(- 5\right) - 12 = 3$

Thus, $z = - x - y - 1 = - 3 - \left(- 5\right) - 1 = 1$