Math ↠ Fractional equations ↠ Fractional equations with several fractions

Fractional equations with several fractions

Solving a fractional equation with multiple fractions is similar to solving fractional equations with just one fraction. Before that, however, the fractions must be brought to a common main denominator.

!

Remember

To bring fractions to a common denominator, multiply the numerator and denominator of a fraction by the denominators of the other fractions.
i

Hint

If there is only one fraction on either side of the equation, it makes sense to multiply the entire equation by the two denominators of the fractions. This breaks up the fraction terms.

Example

Solve the following equation: $\frac{5x}{3x+15}=\frac{5}{6}$

  1. Determine the domain of a function

    $3x+15=0\quad|-15$
    $3x=-15\quad|:3$
    $x=-5$

    $\mathbb{D}=\mathbb{R}\backslash\{-5\}$

  2. Change equation to $x$

    Version 1
    Bring both fractions to a common denominator.

    $\frac{5x}{3x+15}=\frac{5}{6}\quad|-\frac{5}{6}$

    $\frac{5x}{\color{blue}{3x+15}}-\frac{5}{\color{green}{6}}=0$
    $\frac{5x}{\color{blue}{3x+15}}\cdot\frac{\color{green}{6}}{\color{green}{6}}-\frac{5}{\color{green}{6}}\cdot\frac{\color{blue}{3x+15}}{\color{blue}{3x+15}}=0$
    $\frac{30x}{6(3x+15)}-\frac{5\cdot(3x+15)}{6(3x+15)}=0$

    $\frac{30x-5\cdot(3x+15)}{6(3x+15)}=0\quad|\cdot6(3x+15)$

    $\frac{30x-5\cdot(3x+15)}{6(3x+15)}\cdot6(3x+15)=0\cdot6(3x+15)$

    $30x-5\cdot(3x+15)=0$
    $30x-15x-75=0$
    $15x-75=0\quad|+75$
    $15x=75\quad|:15$

    $x=5$


    Version 2 (See hint)
    In order to resolve the fraction terms, the equation is multiplied by the denominators of the fractions.

    $\frac{5x}{3x+15}=\frac{5}{6}\quad|\cdot6\cdot(3x+15)$

    $6\cdot5x=5\cdot(3x+15)$
    $30x=15x+75=0\quad|-15x$
    $15x=75\quad|:15$

    $x=5$

  3. Check result

    Check if the result is included in the domain

    $x=5$ is contained in the domain $\mathbb{D}=\mathbb{R}\backslash\{-5\}$: The solution is valid.