Question

Write an equation that represents the statement. The product of \(x\) and 6 equals \(-9\).

Short answer

The equation that represents the given statement is \(6x = -9\).

Step-by-step solution

Identify the mathematics in the statement

In the statement, there's a reference to a 'product', which refers to the result of multiplying numbers together. The numbers we're multiplying are \(x\) and \(6\). There's also the term 'equals', which in mathematics is represented by the equals symbol =. So we're looking for a product that's equal to \(-9\).

Write the equation

Knowing that the product of \(x\) and \(6\) is \(-9\), the equation is then represented as: \(x \cdot 6 = -9\), because the product of \(x\) and \(6\) equals to \(-9\).

Simplify the Equation

The equation \(x \cdot 6 = -9\) can be simplified to \(6x = -9\).

Key concepts

Product of Numbers

When we talk about the product of numbers, we are referring to the result of a multiplication operation. In the context of our exercise, the numbers are the variable \( x \) and the constant 6. Multiplying \( x \) by 6 gives us what we call the *product*. The term "product" is used to show that two or more numbers have been multiplied together. In our example equation, \( x \cdot 6 = -9 \), the "product" indicates what this multiplication equals. Understanding the product as the outcome of multiplication is crucial for solving equations where multiplication is involved.

Simplification in Algebra

Simplification in algebra is a process where we make an expression or equation as straightforward as possible, without changing its value. It's about reducing complexity while maintaining the same mathematical truth. For our equation, \( x \cdot 6 = -9 \), simplification involves combining the multiplication into a more standard form. Here, this means writing it as \( 6x = -9 \). This is simpler because it follows traditional mathematical notation and makes future operations, like solving for \( x \), easier. Always aim to simplify wherever possible to avoid errors and increase clarity in solving equations. Steps to simplify:

• Combine similar terms or factors.
• Rewrite expressions in standard algebraic form.
• Ensure the simplified expression is equivalent to the original.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operators like addition, subtraction, multiplication, and division. These expressions form the building blocks of equations, which are essential in algebra.In our problem, \( 6x = -9 \) is an example of an algebraic equation. It consists of a term \( 6x \), where 6 is the coefficient and \( x \) is the variable. An algebraic expression like this can represent many things, depending on \( x \), and can be manipulated to solve for \( x \).A few characteristics of algebraic expressions include:

• Consistency in maintaining equal values when simplified.
• Use of variables to represent unknowns or quantities that can change.
• Ability to model numerical relationships effectively.

Understanding how to form and manipulate these expressions is vital for anyone working with algebra.