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


In a previous section, we practiced problems like

$$2x - 3 = 11$$

Now let's make these a little more complicated by introducing more fractions. In this section we'll work problems that look like this:

$$\frac{4}{5}x - \frac{1}{3} = \frac{2}{7}$$

These will give you both some algebra practice and some (always needed) practice adding and multiplying fractions. To solve the equation above, we'd follow these steps. First, we'll want to get the $\frac{1}{3}$ to the right side (away from x)

$$ \begin{align} &\frac{4}{5}x - \frac{1}{3} = \frac{2}{7} \\ &\underline{\phantom{000} +\frac{1}{3} + \frac{1}{3}}\\ &\phantom{0000} \frac{4}{5}x = \frac{1}{3} + \frac{2}{7} = \frac{7 + 6}{21} \\ \end{align}$$

Now we just need to move the $\frac{4}{5}$ over by multiplying both sides of our equation by the reciprocal, $\frac{5}{4}$:

$$ \begin{align} \frac{5}{4}\cdot \frac{4}{5}x &= \frac{13}{21}\cdot \frac{5}{4} \\[5pt] x &= \frac{65}{104} \end{align}$$

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

These are good problems for practicing your fraction skills, too. enter your answers as fractions like #/#. Half an hour of practice will really help you improve those important skills.









In the next algebra practice section we'll work on problems in which the variable, x, is in the denominator of a fraction, like

$$\frac{3}{x} - 2 = 9.$$


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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