Problem 26 - Algebra 2

Question

Write a verbal expression to represent each equation. \(3 a^{3}=a+4\)

Step-by-Step Solution

Answer

Three times \(a\) cubed equals the sum of \(a\) and four.

Step 1: Identify Equation Components

Recognize that the equation is a statement showing equality between two expressions: the left side is \(3a^{3}\) and the right side is \(a+4\).

Step 2: Express the Left Side Verbally

The left side of the equation, \(3a^{3}\), can be expressed verbally as 'three times \(a\) cubed'.

Step 3: Express the Right Side Verbally

The right side of the equation, \(a+4\), can be expressed verbally as 'the sum of \(a\) and four'.

Step 4: Combine for the Complete Expression

Combine the verbal expressions for both sides to describe the whole equation as 'three times \(a\) cubed equals the sum of \(a\) and four.'

Key Concepts

Equation Components

Equations are fundamental in algebra and mathematics as they represent a statement of equality between two expressions. Understanding the components of an equation helps in interpreting and solving it. An equation typically consists of two main parts: the left-hand side and the right-hand side, separated by an equality sign \( (=) \).

• Left-Hand Side (LHS): This is usually the expression or terms written before the equality sign. For instance, in the equation \( 3a^{3} = a + 4 \), the LHS is \( 3a^{3} \).
• Right-Hand Side (RHS): This is the expression or terms written after the equality sign. In our example, the RHS is \( a + 4 \).

Understanding these components is crucial, as it allows you to interpret how one set of terms is related to another. The equality sign indicates that both sides represent the same value, which is the ultimate goal of solving an equation.

Mathematical Expressions

In algebra, mathematical expressions are vital as they express quantities and relationships in a concise form. A mathematical expression can consist of numbers, variables, operations, and sometimes grouping symbols like parentheses.

• In the expression \( 3a^{3} \), we have a constant (3), a variable \( a \), and an exponent (3), indicating that \( a \) is cubed and then multiplied by 3.
• Similarly, \( a + 4 \) is another expression that involves a variable \( a \) and a constant (4), with the operation being addition.

Mathematical expressions do not have equality signs, as they stand alone and are used to form equations. Expressions are the building blocks of equations, and they represent specific calculations or operations to be performed. Understanding how to read and interpret these is key to mastering algebra.

Algebraic Terms

Algebraic terms are the individual parts of a mathematical expression or an equation that are separated by addition or subtraction. Each term can consist of constants, variables, and coefficients. Let’s break down some terms from our example equation, \( 3a^{3} = a + 4 \):

• 3a³: This is a single term that includes a coefficient (3), a variable \( a \), and an exponent (3). The coefficient indicates how many times the term is to be multiplied.
• a: Another term, which is just the variable itself with an implicit coefficient of 1.
• 4: A constant term, which stands alone without any variables.

Terms are crucial because algebraic operations are often performed on them to simplify expressions or solve equations. Understanding terms helps to identify the operations needed to manipulate algebraic expressions effectively.