Problem 47 - Intermediate Algebra

Question

Replace each question mark to make the given statement an application of the indicated property of equality. For example, \(16=\) ? becomes \(16=16\) because of the reflexive property of equality. If \(4=3 x+1\), then \(?=4\) (Symmetric property of equality)

Step-by-Step Solution

Answer

3x + 1

Step 1: Understand the Symmetric Property

The symmetric property of equality states that if \(a = b\), then \(b = a\). Essentially, if one expression equals a second expression, you can swap them and they will still be equal.

Step 2: Identify Given Equality

We are given the equation \(4 = 3x + 1\). According to the symmetric property, we can express this as \(3x + 1 = 4\).

Step 3: Replace the Question Mark

In the statement \(? = 4\), replace the question mark with the left-side expression from the originally given equation. So, \(3x + 1 = 4\).

Step 4: Confirm the Symmetric Property

We have rewritten \(4 = 3x + 1\) as \(3x + 1 = 4\). This confirms we've correctly applied the symmetric property of equality by swapping the sides of the equation.

Key Concepts

Properties of Equality

The properties of equality are foundational concepts in algebra that allow us to manipulate equations while keeping their solutions the same. These properties give us tools to rearrange and solve equations systematically. One of the key properties is the symmetric property of equality.

• Symmetric Property: If \(a = b\), then \(b = a\). This means we can swap the left and right sides of the equation without changing the truth of the statement.
• Reflexive Property: Any value is equal to itself, represented as \(a = a\).
• Transitive Property: If \(a = b\) and \(b = c\), then \(a = c\).

Understanding these properties helps us to confidently and correctly manipulate algebraic expressions in many different mathematical contexts.

Algebraic Equations

Algebraic equations are mathematical statements that show the relationship between different values. They consist of variables and constants connected by operations like addition, subtraction, multiplication, or division. Solving these equations often involves determining the value of the variables.
For example, consider the equation \(4 = 3x + 1\). Here, we have an equation with one variable \(x\). The goal is to isolate \(x\) on one side to find its value.

• An equation maintains its equality as long as you apply the same operation to both sides. This ensures the balance of the equation is not disturbed.
• Using properties of equality, such as the symmetric property, can simplify our task of rearranging equations.

By understanding the structure and components of algebraic equations, you can easily solve them by systematically applying mathematical principles.

Intermediate Algebra Concepts

Intermediate algebra builds on fundamental concepts, introducing more complex operations and relationships between expressions. It's an important stage that bridges basic algebra and more advanced mathematical topics. Some aspects of intermediate algebra involve the application of properties of equality, factoring expressions, and working with quadratic equations.

• Conceptual understanding is crucial. For instance, recognizing how the symmetric property allows you to solve equations more flexibly.
• Practicing the manipulation of equations using different properties can improve your problem-solving skills.
• Intermediate algebra also covers solving systems of equations, quadratic functions, and rational expressions.

Familiarity with these concepts enhances your ability to approach and solve a wide range of mathematical problems, providing a strong foundation for more advanced topics in mathematics.