# eliminate

Eliminate variables from rational equations

## Syntax

``expr = eliminate(eqns,vars)``

## Description

example

````expr = eliminate(eqns,vars)` eliminates the variables `vars` from the rational equations `eqns`. The result is a vector of symbolic expressions that is equal to zero.```

## Examples

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Create two rational equations that contain the variables `x` and `y`.

```syms x y eqns = [x*y/(x-2) + y == 5/(y - x), y-x == 1/(x-1)]```
```eqns =  $\left(\begin{array}{cc}y+\frac{x y}{x-2}=-\frac{5}{x-y}& y-x=\frac{1}{x-1}\end{array}\right)$```

Eliminate the variable `x`. The result is a symbolic expression that is equal to zero.

`expr = eliminate(eqns,x)`
`expr = $\left[6 {y}^{2}-5 y-75\right]$`

Create two polynomial equations that contain the variables `x` and `y`.

```syms x y eqns = [2*x+y == 5; y-x == 1]```
```eqns =  $\left(\begin{array}{c}2 x+y=5\\ y-x=1\end{array}\right)$```

Eliminate the variable `x` from the equations. The result is a symbolic expression that is equal to zero.

`expr = eliminate(eqns,x)`
`expr = $\left[3 y-7\right]$`

Now, create three polynomial equations that contain the variables `x`, `y`, and `z`. Eliminate the variable `x`. The result is a vector of symbolic expressions that is equal to zero.

```syms z eqns = [x^2 + y-z^2 == 2; x - z == y; x^2 + y^2-z == 4]; expr = eliminate(eqns,x)```
`expr = $\left[5 {z}^{3}-5 {z}^{2}-8 z+4 y-8,5 {z}^{4}-11 {z}^{2}-18 z-8\right]$`

To eliminate both `x` and `y`, use the `eliminate` function and specify the two variables as the vector `[x y]`.

`expr = eliminate(eqns,[x y])`
`expr = $\left[5 {z}^{4}-11 {z}^{2}-18 z-8\right]$`

## Input Arguments

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Rational equations, specified as a vector of symbolic equations or symbolic expressions. A rational equation is an equation that contains at least one fraction in which the numerator and the denominator are polynomials.

The relation operator `==` defines symbolic equations. If a symbolic expression `eqn` in `eqns` has no right side, then a symbolic equation with a right side equal to `0` is assumed.

Variables to eliminate, specified as a vector of symbolic variables.