17.3 Multiplicative One-Step Equations

Multiplicative One-Step Equations are equations in which a variable is connected to the remainder of the equation through exactly one multiplication or division operation, and which are solved by applying a single inverse multiplicative operation to both sides. These equations form the multiplicative branch of one-step linear equations, distinct from equations whose sole operation is addition or subtraction.

Integer Coefficient Equation describes the form in which a positive whole-number coefficient multiplies the variable, producing the general structure shown below.

Here, the coefficient a is a positive integer multiplying the variable x, and the product equals a known value b. Recognizing this pattern determines that the coefficient must be removed through division rather than through any additive operation.

Negative Integer Coefficient Equation describes the same multiplicative structure, but with a coefficient that is a negative whole number rather than a positive one. The equation still takes the form of a coefficient multiplying the variable equal to a value, except the coefficient itself carries a negative sign. This case requires the same inverse operation, division, but the sign of the coefficient must be carried through the division correctly, since dividing both sides by a negative number reverses the sign of the resulting solution relative to what a positive coefficient of the same magnitude would produce.

Variable Divided by an Integer describes the form in which the variable itself is divided by a positive whole number, producing the general structure shown below.

In this form, the variable is the numerator of a fraction with a fixed positive integer denominator, and this fraction equals a known value. Removing the division requires multiplying both sides by that same integer denominator.

Variable Divided by a Negative Integer describes the same division structure, but with a negative whole number as the denominator. The inverse operation remains multiplication by the denominator, but because the denominator is negative, the resulting solution carries a sign consistent with multiplying by a negative value, which must not be confused with the sign that would result from a positive denominator of the same magnitude.

Negated Variable Equation addresses the special case in which the variable appears with an implied coefficient of negative one, such as an equation stating that the negative of the variable equals some value. Although no explicit numeral appears, this is still a multiplicative one-step equation with a coefficient of negative one, and it is resolved by dividing both sides by negative one, or equivalently by multiplying both sides by negative one, which reverses the sign of both sides and isolates the variable with a positive coefficient.

Multiplication Inverse Operation Selection applies whenever the variable is divided by a constant, as in Variable Divided by an Integer or Variable Divided by a Negative Integer. Because division and multiplication are inverse operations, the constant denominator is removed by multiplying both sides of the equation by that same constant, invoking the multiplication property of equality.

Division Inverse Operation Selection applies whenever a constant coefficient multiplies the variable, as in Integer Coefficient Equation, Negative Integer Coefficient Equation, or Negated Variable Equation. The coefficient is removed by dividing both sides of the equation by that same nonzero coefficient, invoking the division property of equality. This selection is only valid when the coefficient is nonzero, since division by zero is undefined and would invalidate the equation entirely.

Multiplicative Variable Isolation is the resulting state after the correct inverse operation has been applied: the variable stands alone with a coefficient of exactly one, and the opposite side of the equation contains a single numerical value representing the solution. If any coefficient other than one remains attached to the variable after the operation, the inverse step was performed incorrectly or incompletely.

Multiplicative Equation Solution Check is the verification step performed after isolating the variable, in which the found value is substituted back into the original equation, before any inverse operation was applied, to confirm both sides evaluate to the same quantity. This check carries out the original multiplication or division exactly as it appeared before solving and compares the result to the value on the opposite side. Agreement confirms the solution is correct, while disagreement indicates either an arithmetic mistake or an incorrect choice of inverse operation, requiring the isolation process to be revisited.