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


In this section you can work algebra problems in which the variable, x, is in the denominator of a fraction. It's important to know how to do these. One very clear rule of algebra is

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 such a problem:

$$\frac{3}{x} - 4 = 11$$

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

$$ \begin{align} &\frac{3}{x} - 4 = 11 \\ &\underline{\phantom{00} + 4 \; + 4} \\ &\phantom{000}\frac{3}{x} = 15 \end{align}$$

Now we need to get the x out of the denominator. We do this my multiplying by x on both sides of the equation:

$$3 = 15x$$

Now dividing by 15 gets the x by itself, and we have

$$x = \frac{3}{15} = \bf \frac{1}{5}.$$

Another way to think about that last step is by cross-multiplication. If we write

$$\frac{3}{x} = 15 \; \; \color{#E90F89}{\text{ as }} \; \; \frac{3}{x} = \frac{15}{1},$$

we can cross multiply

to get $15x = 3,$ and so on.

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.









Next we'll do some of these in which the integers are fractions. Everybody needs practice with fractions!


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.

  • Multiplication and division are inverse operations.

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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