# How to Rearranging a Multi-Variable Equation

Expressing one variable in terms of other variables is called rearranging a multivariable equation. You can rearrange multivariable equations by following this article’s step-by-step guide.

## A step-by-step guide to Rearranging a Multi-Variable Equation

Expressing one variable in terms of other variables is called rearranging a multivariable equation. The process of rearranging the multivariable equation is not much different from the single-variable equation, only more expressions and variables are involved. You can rearrange multivariable equations by following the step-by-step guide below:

Step $$1$$: Isolating the variable: First, you need to isolate the variable that you want to express in terms of the others. To isolate, you can use expression addition or subtraction, multiplication or division of both sides by a constant, or inversion operations.

Step $$2$$: Simplify the equation: Use the distributive property or power properties, logarithms, or trigonometric functions to simplify the equation.

Step $$3$$: Simplify the equation further: use algebraic operations such as factoring and expansion

Step $$4$$: Do the inverse operation: Use the inverse operation, such as taking the reciprocal or square root, to cancel the expression.

Step $$5$$: Checking the correctness of the operation: check the correctness of your work by inserting the obtained values into the main equation.

Step $$6$$: Write the final answer: Write your answer in a complete sentence.

The rearranging process varies from one equation to another according to the complexity of the equation and depends on the form of the equation and the variables involved.

Note: For some equations, such as equations that are non-linear or those that include non-algebraic functions, the rearranging process is impossible.

### Rearranging a Multi-Variable Equation-Example 1:

Rearrange the equation $$4x+5y=20$$ for $$y$$ in terms of $$x$$.

Solution:

To rearrange this equation for $$y$$ in terms of $$x$$, we can start by isolating $$y$$ on one side of the equation. We can do this by subtracting $$4x$$ from both sides:

$$4x+5y=20$$$$4x+5y-4x=20-4x$$$$5y=20-4x$$

Next, we can divide both sides by $$5$$ to get $$y$$ alone:

$$\frac{5y}{5}=\frac{20-4x}{5}$$

So we have $$y=\frac{20-4x}{5}$$

### Rearranging a Multi-Variable Equation-Example 2:

Rearrange the equation $$5x^2+6y^2 =25$$ for $$x$$ in terms of $$y$$.

Solution:

To rearrange this equation for $$x$$ in terms of $$y$$, we can start by isolating $$x^2$$ on one side of the equation. We can do this by subtracting $$6y^2$$ from both sides:

$$5x^2+6y^2 =25$$$$5x^2+6y^2-6y^2=25-6y^2$$$$5x^2=25-6y^2$$

$$\frac{5x^2}{5}=\frac{25-6y^2}{5}$$$$x^2=\frac{25-6y^2}{5}$$

Next, we can take the square root of both sides to get $$x$$ alone: $$x=\sqrt{\frac{16 – 2y^2}{4}}$$

So we have$$x=\sqrt{\frac{16 – 2y^2}{4}}$$

In this way, we have rearranged the equation for $$x$$ in terms of $$y$$, and we can use this expression to find the value of $$x$$ for a given value of $$y$$.

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