1Cademy - Classifying $$6(2n - 1) + 3 = 2n

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Classification of Linear Equations

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Classifying $6(2n - 1) + 3 = 2n - 8 + 5(2n + 1)$

To determine whether $6(2n - 1) + 3 = 2n - 8 + 5(2n + 1)$ is a conditional equation, an identity, or a contradiction, simplify both sides. Distributing yields $12n - 6 + 3 = 2n - 8 + 10n + 5$ . Combining like terms simplifies the given equation down to $12n - 3 = 12n - 3$ . Subtracting $12n$ from each side to isolate the variable removes $n$ entirely, leaving the true constant statement $-3 = -3$ . Because this statement is inherently true independent of the value of $n$ , the equation is classified as an identity and its solution is all real numbers.

Updated 2026-06-18

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OpenStax Intermediate Algebra

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An operations analyst is comparing two different labor-hour formulas represented by the equation ${}6(2n - 1) + 3 = 2n - 8 + 5(2n + 1)$ , where $n$ is the number of staff members. After simplifying both sides of the equation, the analyst arrives at the true statement ${}-3 = -3$ . What is the mathematical classification for this type of equation?
A telecommunications engineer uses the equation ${}6(2n - 1) + 3 = 2n - 8 + 5(2n + 1)$ to compare two signal strength models, where $n$ is the number of repeaters. After applying the distributive property and combining like terms, the engineer simplifies the equation to the true statement ${}-3 = -3$ . In mathematics, this type of equation that is true for all possible values of the variable is called a(n) ____.
An office manager is evaluating two different monthly maintenance cost models for a fleet of vehicles. The relationship between the models is represented by the equation ${}6(2n - 1) + 3 = 2n - 8 + 5(2n + 1)$ , where $n$ is the number of vehicles. Arrange the following steps in the correct order to simplify and classify the relationship between these two models.
A retail manager uses the equation ${}6(2n - 1) + 3 = 2n - 8 + 5(2n + 1)$ to verify that two different discount strategies will always result in the same final price for any number of items ordered, $n$ . After simplifying the equation to the true statement ${}-3 = -3$ , the manager correctly concludes that the equation is an identity. True or False?
A warehouse logistics coordinator is comparing two automated crate-tracking models to see if they always yield the same result. The relationship is represented by the equation ${}6(2n - 1) + 3 = 2n - 8 + 5(2n + 1)$ , where $n$ represents the number of crates. Match each mathematical term or stage of the simplification process with its correct value or description.
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