Problem 92 - Beginning and Intermediate Algebra

Question

Determine whether each is an expression or an equation. a. \(3 x^{2}-26\) b. \(3 x^{2}-26=1\) c. \(2 x-5=7 x-5\) d. \(9 y+x-8\)

Step-by-Step Solution

Answer

a: Expression, b: Equation, c: Equation, d: Expression

Step 1: Understand Expressions

An expression is a combination of numbers, variables, and operators (such as +, −, ×, ÷) without an equality sign (=). It is like a mathematical phrase that can be simplified but not solved, as it doesn't equate to anything.

Step 2: Understand Equations

An equation is a mathematical statement that asserts the equality of two expressions, connected by an equality sign (=). It is like a complete mathematical sentence that can be solved to find the value of the variables involved.

Step 3: Analyze Each Option - Part (a)

The given part (a) is \(3x^2 - 26\). This expression consists of a polynomial with a single variable term \(3x^2\) and a constant \(-26\). There is no equality sign, so it is an expression.

Step 4: Analyze Each Option - Part (b)

The given part (b) is \(3x^2 - 26 = 1\). This includes an expression \(3x^2 - 26\) equated to a number 1. Since there's an equality sign linking two expressions, it is an equation.

Step 5: Analyze Each Option - Part (c)

The given part (c) is \(2x - 5 = 7x - 5\). Here, two expressions \(2x - 5\) and \(7x - 5\) are equated, as it contains an equality sign. Therefore, it is an equation.

Step 6: Analyze Each Option - Part (d)

The given part (d) is \(9y + x - 8\). This expression has terms \(9y\), \(x\), and a constant \(-8\), without an equality sign. Consequently, it is an expression.

Key Concepts

Equations

In the world of mathematics, understanding the concept of equations is essential. Equations are powerful tools because they assert that two things are equal. An equation has an equality sign "=" which separates two expressions or terms. It's like a mathematical scale, where both sides must equal each other. For example, in the exercise question, parts (b) and (c) are equations.

• Part (b): \(3x^2 - 26 = 1\) shows an expression \(3x^2 - 26\) equated to 1.
• Part (c): \(2x - 5 = 7x - 5\) equates two different expressions together.

Equations are not just for showing equality, but they are tools for problem-solving. By manipulating equations, you can isolate variables and find their values. In algebra, this process of finding the value that makes the equation true is essential.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operators, such as addition, subtraction, multiplication, and division. These expressions do not include an equality sign "=". They represent quantities and can be evaluated or simplified but not solved, as they do not equate to anything. In the original exercise, examples of expressions are:

• Part (a): \(3x^2 - 26\), which includes a variable term and a constant.
• Part (d): \(9y + x - 8\), which has terms with variables and a constant.

Expressions are everywhere in mathematics, from simple arithmetic to complex algebra. They help to form the components of equations and do not have to "solve" anything. Their primary role is to represent values and relationships.

Polynomial Expressions

Another key concept is polynomial expressions, which are specific types of algebraic expressions. A polynomial consists of terms that are made up of coefficients (numbers) and variables raised to whole number powers. It can have one or many terms. Each term in a polynomial is separated by plus or minus signs.In the exercise, we can identify that parts (a) and (b) include polynomial expressions:

• Part (a): \(3x^2 - 26\) is a polynomial expression with a single variable term \(3x^2\).
• Part (b): Within the equation \(3x^2 - 26 = 1\), the expression \(3x^2 - 26\) is a polynomial.

Polynomials are incredibly useful in algebra—they can represent a wide range of real-world situations, from physics to economics. They can be added, subtracted, multiplied, and even divided (under certain conditions), making them versatile in operations.