Question

Tell whether each rule is \(\bullet\) a convention, or \(\bullet\) a rule we can't change Nine times a number is equal to the difference between 10 times the number and the number.

Short answer

It is a rule we can't change.

Step-by-step solution

Understand the Problem

We need to classify a given rule as either a convention or a rule we can't change. The mathematical statement is: Nine times a number is equal to the difference between 10 times the number and the number.

Translate the Statement into an Equation

Translate the given sentence into a mathematical equation. Let the number be represented as x. The equation becomes: \( 9x = 10x - x \).

Simplify the Equation

Simplify the equation to check its correctness. Starting with \( 9x = 10x - x \), notice that the right-hand side simplifies to \( 9x \). Thus, \( 9x = 9x \), which is always true for any value of x.

Determine the Type of Rule

Since the equation \( 9x = 10x - x \) is derived from basic algebraic principles, it is a fundamental property based on the axioms of arithmetic and algebra. Thus, it is a rule we can't change.

Key concepts

algebraic equations

An algebraic equation is a mathematical statement that shows the equality of two expressions by connecting them with an equal sign (\texttt{=}).
For example, in the exercise, the statement 'Nine times a number is equal to the difference between 10 times the number and the number' can be translated into an algebraic equation as follows: Let \( x \) be the number, then the equation is \( 9x = 10x - x \). This type of equation helps us understand the relationships between numbers and their operations.
Equations can be simple (linear) or complex (quadratic, cubic, etc.), depending on the degree and number of variables involved.
Solving algebraic equations typically involves isolating the variable on one side of the equation. In our example, simplifying \( 10x - x \) gives us \( 9x \), resulting in \( 9x = 9x \), which shows that the equation holds true for any value of \( x \).

mathematical conventions

Mathematical conventions are agreed-upon norms and notations used in mathematics to maintain consistency and clarity.
Some common examples include the order of operations (PEMDAS/BODMAS) and the use of specific symbols such as \( + \) for addition, \( - \) for subtraction, and \( \cdot \) or \( \times \) for multiplication.
These conventions help ensure everyone interprets mathematical expressions and equations the same way.
The rule given in the exercise, 'Nine times a number is equal to the difference between 10 times the number and the number,' follows the standard convention where expressions are written sequentially for clarity and simplicity.
However, the specific form of this rule, such as \( 9x = 10x - x \), is derived from fundamental principles rather than being a mere convention or notation.

arithmetic axioms

Arithmetic axioms are fundamental truths in mathematics that serve as the basis for deriving other truths and principles.
They are self-evident and universally accepted without proof. Some of the basic axioms include the Commutative, Associative, and Distributive properties.
For example, the Commutative Property of Addition states that \( a + b = b + a \). Similarly, the Distributive Property states that \( a(b + c) = ab + ac \).
In the exercise, the equation \( 9x = 10x - x \) simplifies to \( 9x = 9x \), which is primarily based on the arithmetic axiom that allows us to perform and simplify operations on both sides of the equation.
This outcome reflects the inherent consistency and reliability of these axioms in solving mathematical problems.