Algebraic Fractions (Equations)

Vocabulary and procedural flashcards covering methods, definitions, and worked examples for solving linear and quadratic algebraic fraction equations.

Algebraic Fraction

A mathematical fraction in which the numerator, denominator, or both contain an algebraic term or variable expression (e.g., $\frac{x+9}{5}$ or $\frac{1}{x+3}$).

Lowest Common Denominator (LCD)

The simplest algebraic expression or smallest number that is a common multiple of all denominators in an equation, used to eliminate fractions by multiplying every term on both sides.

Clearing Fractions

The algebraic procedure of multiplying every term on both sides of an equation by the common denominator to remove all fraction bars and simplify the equation.

Linear Algebraic Fraction Equation

An equation containing algebraic fractions with purely numerical denominators that, once cleared of denominators, directly reduces to a first-degree equation in the form $ax + b = c$.

Quadratic Algebraic Fraction Equation

An equation containing algebraic fractions with variable binomial expressions in the denominators that, when cleared and expanded, simplifies into a second-degree polynomial equation of the form $ax^2 + bx + c = 0$.

Sign Distribution in Algebraic Fraction Subtraction

The rule requiring that when subtracting an algebraic fraction with a multi-term numerator, the negative sign applies to every term in that numerator upon expansion, such as $-(x - 2) = -x + 2$.

Exact Surd Form ($\frac{a \pm \sqrt{b}}{c}$)

The closed-form representation of the roots of a quadratic equation using integers $a$, $b$, and $c$, where irrational roots are retained in radical form rather than converted to rounded decimals.

Solving Equation 1 ($\frac{x+9}{5} + \frac{x+2}{4} = 5$)

Multiply all terms by the common denominator $20$ to yield $4(x+9) + 5(x+2) = 100$, simplify to $9x + 46 = 100$, and solve to obtain $x = 6$.

Solving Equation 2 ($\frac{x-1}{2} + \frac{x+4}{5} = 8$)

Multiply all terms by the common denominator $10$ to obtain $5(x-1) + 2(x+4) = 80$, simplify to $7x + 3 = 80$, and solve to obtain $x = 11$.

Solving Equation 3 ($\frac{x+5}{3} - \frac{x-2}{4} = 3$)

Multiply all terms by the common denominator $12$ to obtain $4(x+5) - 3(x-2) = 36$, expand carefully to $4x + 20 - 3x + 6 = 36$, and simplify to find $x = 10$.

Solving Equation 4 ($\frac{x+2}{8} + \frac{5-x}{3} = 2$)

Multiply all terms by the common denominator $24$ to obtain $3(x+2) + 8(5-x) = 48$, simplify to $46 - 5x = 48$, and solve to get $x = -\frac{2}{5}$ (or $-0.4$).

Question 12 Problem Type ($\frac{x}{x+1} + \frac{1}{x-2} = -2$)

A 6-mark algebraic fraction problem requiring multiplying by $(x+1)(x-2)$, rearranging into a quadratic equation, and applying the quadratic formula to yield answers in the exact format $\frac{a \pm \sqrt{b}}{c}$.