Calculatrice Equations rationnelles

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 Exemple

 Problèmes résolus

 Problèmes difficiles

Here, we show you a step-by-step solved example of rational equations. This solution was automatically generated by our smart calculator:

$\frac{2}{x+1}=\frac{3}{x-1}$

Inverting the equation

$\frac{x+1}{2}=\frac{x-1}{3}$

Expand the fraction $\frac{x+1}{2}$ into $2$ simpler fractions with common denominator $2$

$\frac{x}{2}+\frac{1}{2}=\frac{x-1}{3}$

Multiply both sides of the equation by $3$

$x-1=3\left(\frac{x}{2}+\frac{1}{2}\right)$

Solve the product $3\left(\frac{x}{2}+\frac{1}{2}\right)$

$x-1=3\left(\frac{x}{2}\right)+3\cdot \left(\frac{1}{2}\right)$

Multiplying the fraction by $3$

$x-1=\frac{3x}{2}+3\cdot \left(\frac{1}{2}\right)$

Multiply the fraction and term in $3\cdot \left(\frac{1}{2}\right)$

$x-1=\frac{3x}{2}+\frac{3}{2}$

Solve the product $3\left(\frac{x}{2}+\frac{1}{2}\right)$

$x-1=\frac{3x}{2}+\frac{3}{2}$

Combine fractions with common denominator $2$

$x-1=\frac{3x+3}{2}$

Move everything to the left hand side of the equation

$x-1+\frac{-3x-3}{2}=0$

Combine all terms into a single fraction with $2$ as common denominator

$\frac{2x+2\cdot -1-3x-3}{2}=0$

Multiply $2$ times $-1$

$\frac{2x-2-3x-3}{2}=0$

Combine all terms into a single fraction with $2$ as common denominator

$\frac{2x-2-3x-3}{2}=0$

Subtract the values $-2$ and $-3$

$\frac{2x-5-3x}{2}=0$

Combining like terms $2x$ and $-3x$

$\frac{-x-5}{2}=0$

Multiply both sides of the equation by $2$

$-x-5=0\cdot 2$

Multiply $0$ times $2$

$-x-5=0$

Multiply both sides of the equation by $2$

$-x-5=0$

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-5$ from both sides of the equation

$-x-5+5=0+5$

Canceling terms on both sides

$-x=5$

Multiply both sides of the equation by $-1$

$x=-5$

Verify that the solutions obtained are valid in the initial equation

The valid solutions to the equation are the ones that, when replaced in the original equation, don't make any denominator equal to $0$, since division by zero is not allowed

The equation has no solutions.

 Réponse finale au problème