Problem 45 - 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 \(n=2\) and \(3 n+4=10\), then \(3(?)+4=10\) (Substitution property of equality)

Step-by-Step Solution

Answer

Replace \(n\) with 2 to get \(3(2) + 4 = 10\).

Step 1: Understanding the Substitution Property

The substitution property of equality states that if two things are equal, one can be replaced by the other in an expression or equation. This means if we have an equation where a part equals 'n' and we know 'n' equals 2, then '2' can substitute 'n' in any expression.

Step 2: Identify the Expression to Substitute

In the expression given, \(3n + 4 = 10\), we need to identify where we will substitute the value \(n = 2\). The \(n\) in the expression \(3n\) can be replaced with 2 because we know from the information given that \(n = 2\).

Step 3: Perform the Substitution

Now that we know \(n = 2\), substitute 2 in place of \(n\) in the expression \(3n\). Therefore, the expression becomes \(3(2) + 4 = 10\).

Step 4: Verify the Result

Check if the new expression is valid: \(3(2) + 4 = 6 + 4 = 10\), which holds true, thus confirming that the substitution was correctly done.

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental component of algebra. They consist of numbers, variables, and mathematical operations such as addition, multiplication, etc. For instance, the expression \(3n + 4\) includes a variable \(n\), a numerical coefficient \(3\), the constant \(4\), and the operations of multiplication and addition.

Variables in algebraic expressions act as placeholders, which can take on different values. They can be substituted with known values to solve problems. Understanding how to manipulate and work with algebraic expressions is essential for solving equations correctly. It helps in simplifying expressions, solving equations, and finding solutions to real-world problems.

Equality Properties

Equality properties are the rules that apply when dealing with equations, which are statements that show two expressions are equal. These properties help us understand how equations work and provide methods to manipulate and solve them effectively.

Some common equality properties include:

• Reflexive Property: Anything is equal to itself, \(a = a\).
• Symmetric Property: If \(a = b\), then \(b = a\).
• Transitive Property: If \(a = b\) and \(b = c\), then \(a = c\).
• Substitution Property: If \(a = b\), \(a\) can be substituted for \(b\) in any expression involving \(a\).

Understanding these properties, especially the substitution property, is crucial when solving equations as it allows for replacement of equivalent terms, facilitating solution strategies to algebraic problems.

Equation Solving

Equation solving involves finding the value of variables that make an equation true. This process usually relies on manipulating the equation using algebraic techniques and applying rules like substitution and balancing.

By substituting known values into an equation, as demonstrated in the original exercise, we can transform and simplify the equation. For example, replacing \(n = 2\) into \(3n + 4\) resulted in \(3(2) + 4 = 10\). Such substitution reveals the truth of the equation and helps verify if our solution is correct.

Effective equation solving demands a step-by-step approach:

• Identify what needs to be solved for.
• Use the known values and appropriate equality properties to simplify the equation.
• Perform algebraic operations to isolate the variable.
• Verify the solution by substituting back to check if the original equation holds true.

Mastery of these steps empowers students to solve equations with confidence and precision.