# How does one go about using the elimination method to solve an equation?

## I would love a step by step process please! I just can't seem to understand it the way i was taught. Thank you!!! Here is an example: Solve the system using the elimination method. x+y−2z=5 −x+2y+z=2 2x+3y−z=9

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Feb 9, 2018

$x = 1$, $y = 2$ and $z = - 1$

#### Explanation:

Given the system:

$\left\{\begin{matrix}x + y - 2 z = 5 \\ - x + 2 y + z = 2 \\ 2 x + 3 y - z = 9\end{matrix}\right.$

We can eliminate $z$ from the first equation by adding twice the second equation to it to get:

$- x + 5 y = 9$

We can eliminate $z$ from the second equation by adding the third equation to it to get:

$x + 5 y = 11$

We can eliminate $y$ from this last equation by subtracting the previous equation to get:

$2 x = 2$

Then we can divide both sides by $2$ to find:

$\textcolor{b l u e}{x = 1}$

We can substitute this value of $x$ into the previous equation to find:

$1 + 5 y = 11$

Subtracting $1$ from both sides, we get:

$5 y = 10$

Then dividing both sides by $5$ we find:

$\textcolor{b l u e}{y = 2}$

Substituting these values for $x$ and $y$ into the second equation we were given, we find:

$- \left(1\right) + 2 \left(2\right) + z = 2$

That is:

$3 + z = 2$

Subtracting $3$ from both sides, we find:

$\textcolor{b l u e}{z = - 1}$

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