Problem 43 - 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 \(y=x\) and \(x=-6\), then \(y=\) ? (Transitive property of equality)

Step-by-Step Solution

Answer

If \(y=x\) and \(x=-6\), then \(y=-6\).

Step 1: Understand the Transitive Property of Equality

The transitive property of equality states that if one quantity equals a second quantity and that second quantity equals a third quantity, then the first quantity equals the third quantity. Mathematically, if \(a = b\) and \(b = c\), then \(a = c\).

Step 2: Apply the Transitive Property to the Given Equation

According to the problem statement, you know two things: \(y = x\) and \(x = -6\). You need to determine the value of \(y\).

Step 3: Replace \(x\) with \(-6\) in terms of \(y\)

Since \(y = x\) and \(x = -6\), by substituting \(x\) in the equation for \(y\), we have \(y = -6\). This is consistent with the transitive property of equality.

Key Concepts

Equality in Algebra

In algebra, understanding the concept of equality is fundamental. Equality means that two expressions represent the same value. For example, the equation \(x = y\) implies that whatever value \(x\) holds, \(y\) will also hold that exact value.

There are several properties of equality in algebra that you need to be familiar with:

• Reflexive Property: Any quantity is equal to itself, for example, \(a = a\).
• Symmetric Property: If one quantity equals a second quantity, then the second quantity equals the first, expressed as \(a = b\) implies \(b = a\).
• Transitive Property: This property is crucial in the context of problem-solving. It means if \(a = b\) and \(b = c\), then \(a = c\).

The use of the transitive property was highlighted in the exercise. Understanding these properties is vital to solve algebraic equations precisely.

Problem Solving in Algebra

Algebra problem-solving often involves analyzing given equations and using properties of equality. Here, we have a scenario where we must apply logical reasoning to deduce the value of a variable.
In the exercise, we were given two equations:

• \(y = x\)
• \(x = -6\)

These equations help set the foundation for applying the right algebraic property. It's crucial to approach algebraic problems step-by-step:

• Understand the problem: Clearly identify your variables and what is known or unknown.
• Set up the relationships: Use the given statements to set equations.
• Apply algebraic properties: Use properties like transitive, reflexive, or symmetric to find the solution.
• Verify your answer: Double-check by substituting back to ensure the solution is correct.

Through such systematic approaches, problem-solving in algebra becomes easier to handle and reliable for deriving correct solutions.

Substitution Method

The substitution method is a handy tool in algebra, especially when dealing with equations involving multiple variables. It allows you to replace a variable with its equivalent expression or numerical value to simplify the equation.
In the given problem, substitution plays a critical role. We start with:

• \(y = x\)
• \(x = -6\)

Substitution requires you to take one of these equations and plug it into the other. Since we know \(x = -6\), we can substitute \(-6\) for \(x\) in the equation \(y = x\). This substitution straightforwardly gives us \(y = -6\).

Here's how substitution helps:

• It simplifies complex equations: Reducing the number of variables makes the problem manageable.
• Immediate solutions for variables: Quickly yields values for unknowns.
• Supports other algebraic techniques: Can be combined with elimination and other methods for solving systems of equations.

Utilizing the substitution method is essential for efficient and effective algebraic problem solving.