Problem 9 - Algebra 1: Concepts and Skills

Question

State the meaning of the variable expression and name the operation. $$ (8)(x) $$

Step-by-Step Solution

Answer

In the expression \( (8)(x) \), 'x' is the variable and '8' is the constant. The operation indicated here is multiplication.

Step 1: Identify the Variable

In the expression, \( (8)(x) \), 'x' is the variable. Variables are symbols used to represent unknown values.

Step 2: Identify the Constant

In the expression, \( (8)(x) \), '8' is the constant. Constants are values that don't change.

Step 3: Name the Operation

The two numbers, '8' and 'x', are enclosed in parentheses and stand next to each other. This suggests multiplication, since multiplication in algebra can be indicated by placing numbers and variables next to each other, whether they are in parentheses or not. The operation is therefore multiplication.

Key Concepts

Variables in Algebra

In algebra, a variable is a symbol used to represent a number or value that is not yet known, or can change. The variable acts like a placeholder in expressions and equations.
The most commonly used symbols for variables are letters such as \( x \), \( y \), and \( z \). In the expression \((8)(x)\), the letter \( x \) is the variable. This means \( x \) can represent any number.
Variables allow mathematicians to generalize problems and solve equations, making them a fundamental part of algebra. Here are some key points about variables:

• Variables can take on different values, unlike constants which have a fixed value.
• The value of a variable is often determined by solving the equation or expression it is part of.
• They are used in mathematical formulas and functions to describe relationships between quantities.

Remember, when you see a variable in an equation or expression, think of it as a box awaiting to be filled by a number.

Constants in Algebra

Constants are numbers that remain the same throughout mathematical calculations and do not change, unlike variables. In the expression \((8)(x)\), the number \( 8 \) is a constant.
Since constants hold the same value in any part of the mathematical process, they provide stability within expressions and equations.
Some important points to keep in mind about constants are:

• Constants are essential in defining the scope or limitations in mathematical problems.
• They anchor equations and help simplify the expressions by contributing fixed values.
• Understanding what part of the expression is constant makes it easier to manipulate and solve algebraic problems.

By recognizing constants, you can easily identify which parts of an expression depend on the values of other changing numbers.

Multiplication in Algebra

Multiplication in algebra involves calculating the product of numbers, variables, or both. It is often used to simplify expressions or solve equations.
In some equations, multiplication is shown by putting numbers and variables next to each other. This is what you see in the expression \((8)(x)\).
Here are some facts about multiplication in algebra:

• Placing a number next to a variable, without a symbol, implies multiplication. For example, \((8)(x)\) means \(8 \times x\).
• When variables are multiplied together, the result is typically expressed using an exponent, like \(x \times x = x^2\).
• Multiplication follows the commutative property, meaning \(a \times b = b \times a\).

Recognizing multiplication in expressions helps in evaluating or simplifying them, thus making problem-solving more streamlined.