1.20 Fractional and Decimal Equation Definitions

Fractional and Decimal Equation Definitions establishes the vocabulary for equations whose terms are written as fractions or decimals rather than whole-number coefficients, along with the preparatory techniques used to remove those fractional or decimal forms so that the equation can be solved using the same integer-based isolation sequence applied to simpler equations.

Fractional Equation

A fractional equation is an equation containing one or more terms expressed as fractions, where a variable or constant appears with a nonzero denominator, such as x/3 + 1/2 = 5/6. Fractional equations remain valid algebraic equations and can technically be solved by working directly with the fractions, but doing so is typically more error-prone than first removing the fractions entirely.

Decimal Equation

A decimal equation is an equation containing one or more terms expressed as decimal numbers rather than whole-number coefficients or constants, such as 0.25x + 1.5 = 3.75. Like fractional equations, decimal equations can be solved directly, but converting them to whole-number form first generally simplifies the arithmetic involved in isolating the variable.

Denominator Clearing

Denominator clearing is the technique of multiplying every term on both sides of a fractional equation by the least common denominator of all the fractions present, which cancels each denominator and converts the entire equation into an equivalent equation containing only whole-number coefficients and constants. Applying denominator clearing to the equation above, using the least common denominator of 6, produces 2x + 3 = 5, an equivalent equation free of fractions.

Decimal Clearing

Decimal clearing is the technique of multiplying every term on both sides of a decimal equation by an appropriate power of 10, chosen based on the greatest number of decimal places present among the terms, which shifts every decimal point far enough to the right to eliminate the decimals and convert the equation into an equivalent equation containing only whole-number coefficients and constants. Applying decimal clearing to the equation above, multiplying every term by 100, produces 25x + 150 = 375, an equivalent equation free of decimals.

Together, these definitions describe fractional and decimal equations as ordinary equations wearing an alternate numerical form, and establish denominator clearing and decimal clearing as the standard preparatory transformations that convert either form into an equivalent whole-number equation, after which the familiar isolation sequence can be applied without the added burden of fractional or decimal arithmetic.