xaktly | Algebra | $\frac{a}{x} + \frac{b}{c} = \frac{d}{e}$

Algebra practice 4

Solving problems of the type   $\frac{a}{x} + \frac{b}{c} = \frac{d}{e}$

In this section, we build on the previous section in which we solved problems of the type

$$\frac{a}{x} + b = c,$$

but we replace b and c with fractions. It's important to know how to work with fractions in algebra problems. If you know them well, you'll get through these fine. If you need some practice with fractions and you understood the last set of problems, these will be good exercise.

Here's a reminder about algebra problems in which the variable is in the denominator of a fraction:

If the variable you're trying to isolate is in the denominator, you have to get it out of the denominator.

Here's an example of this class of problem:

$$\frac{1}{x} - \frac{4}{7} = \frac{-3}{2}$$

As usual, the first step to solving such a problem is to do the easy stuff first, namely, move the $\frac{-4}{7}$ to the right by adding it to both sides:

$$ \begin{align} &\frac{1}{x} - \frac{4}{7} = \frac{-3}{2} \\ &\underline{\phantom{00} + \frac{4}{7} \; + \frac{4}{7}} \\ &\phantom{0000}\frac{1}{x} = \frac{-3}{2} + \frac{4}{7} \end{align}$$

Now we can find a common denominator and add the fractions on the right:

$$\frac{1}{x} = \frac{-21 + 8}{14} = \frac{-13}{14}$$

Cross multiplication gives

$$ \begin{align} -13x &= 14 \\[5pt] \text{so } \; \; x &= \frac{-14}{13} \end{align}$$

You can practice problems of this type below. Follow the steps in the example and enter your answer as an integer or a fraction of integers, like 3/4, or a decimal number like 1.55 (if you must). Do as many problems as you need to get good at these. You'll form a solid foundation for what comes next.

Inverse operations

In this section, we'll refer often to inverse operations. Inverse operations are opposite, and one can be used to undo the action of the other.

  • Addition and subtraction are inverse operations. If a smaller common denominator isn't available, remember that

    $$\frac{a}{b} ± \frac{c}{d} = \frac{ad ± bc}{bd}$$

  • Multiplication and division are inverse operations. Remember that division by a fraction is the same as multiplication by its reciprocal:

$$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} \phantom{000} \color{#E90F89}{\text{ or }} \phantom{000} \frac{a}{\frac{b}{c}} = \frac{a\cdot c}{b}$$

Other Algebra practice problems

There are a number of these pages you can use for algebra practice. Just pick the rough type of problem you need to work on.

Type 1:   $ax + by = c$

Type 2:   $\frac{a}{b}x + \frac{c}{d} = \frac{e}{f}$

Type 3:   $\frac{a}{x} + b = c$

Type 4:   $\frac{a}{x} + \frac{b}{c} = \frac{d}{e}$

Type 5:   $ax^2 + b = c$

Type 6:   $ax^2 + \frac{b}{c} = \frac{d}{e}$

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