Fractional First-Degree Equations

A Fractional First-Degree Equation is an equation containing algebraic fractions in which the unknown appears in the denominator and that can be reduced to integer first-degree equations.

To solve such a fractional equation we use the same rules derived from the principles of equivalence but with one extra precaution. We must ensure that the denominators containing the unknown are non-zero. For this reason we must impose Existence Conditions during the solution process.

In this lesson we will examine the solution scheme that lets us find the solutions of fractional first-degree equations.

Fractional First-Degree Equations

From the introductory lesson on equations we know that there are integer equations and fractional equations.

An equation is integer if the unknown never appears in a denominator. Conversely, in a fractional equation the unknown appears at least once in a denominator.

Here we focus on one-variable fractional equations of first degree—i.e. fractional equations that can be reduced to first-degree equations.

Examples of fractional first-degree equations:

Existence Conditions

In general, the first step in solving a fractional first-degree equation is to remove the unknown from the denominators.

Consider

To clear the denominator we apply the second principle of equivalence, multiplying both sides by the polynomial :

We could now simplify the fraction in the left member, but only if we require that the polynomial be non-zero:

If were zero, the division would be meaningless:

Thus the unknown must never take the value :

If a candidate solution violates this condition, the equation is impossible. Hence the condition is called an existence condition.

Continuing the solution, having imposed the condition we may simplify:

We now have an integer first-degree equation:

Solving it as a standard first-degree equation:

Finally, check the solution against the existence condition: since , the solution is valid and the equation is solvable.

Solving a Fractional First-Degree Equation

Once the existence conditions are set, bring all algebraic fractions to a common denominator so the equation can be rewritten as an integer equation.

Consider the more complex example

Here we have three fractions and three denominators containing the unknown.

First, find the existence conditions by setting each denominator non-zero:

Hence

Now move every fraction to the common denominator :

Because each fraction has the same denominator, apply the integer-coefficient reduction rule to eliminate the denominators:

Expand and collect like terms:

Use the cancellation rule to remove :

This is in the form with , so a solution exists:

Finally, verify the solution against the existence conditions:

Because is neither −2 nor 7, it is admissible. The equation is therefore determinate.

Solution Scheme

Summarising, the solution scheme for a fractional first-degree equation is:

In summary

In this lesson we studied fractional first-degree equations, i.e. equations containing algebraic fractions in which the unknown appears in at least one denominator. They are first degree because they can be reduced to integer first-degree equations.

To solve them we impose existence conditions so denominators are non-zero, then convert to a common denominator and reduce to an integer equation. After solving, we must verify that the solution respects the existence conditions; if not, the equation is impossible.