Problem 37 - Prealgebra

Question

OPEN ENDED Give an example of an equation that has no solution and an equation that is an identity.

Step-by-Step Solution

Answer

An equation with no solution is \( x + 3 = x - 2 \); an equation that is an identity is \( 2(x + 3) = 2x + 6 \).

Step 1: Understand the Problem

We need to find two examples. First, an equation that has no solution, meaning no value of the variable makes the equation true. Second, an equation that is an identity, meaning all values of the variable make the equation true.

Step 2: Identify an Equation with No Solution

Consider the equation \( x + 3 = x - 2 \). Subtract \( x \) from both sides to get \( 3 = -2 \). This statement is false, as 3 is never equal to -2. Thus, there are no values of \( x \) that satisfy this equation, making it have no solution.

Step 3: Identify an Equation that is an Identity

Consider the equation \( 2(x + 3) = 2x + 6 \). Expand the left side to get \( 2x + 6 = 2x + 6 \). This statement is always true for all values of \( x \), as both sides are equivalent. Therefore, this equation is an identity.

Key Concepts

No Solution

An equation that has no solution arises when it is impossible to satisfy the equation with any value for the variable. Imagine you're trying to solve an equation like \( x + 3 = x - 2 \). You start by performing operations to determine the value of \( x \). By subtracting \( x \) from both sides, you end up with \( 3 = -2 \), which is a contradiction and clearly not possible. Here, no value for \( x \) will make the equation true. This is a classic case of an equation with no solution, where the variables cancel out to leave an untrue statement. Such situations often arise when we have the same amount of a variable on either side of the equation but different constant terms that do not equate.

Identity Equation

An identity equation is an equation that holds true for all possible values of the variable. For example, consider the equation \( 2(x + 3) = 2x + 6 \). If we expand the left side, we get \( 2x + 6 \), which matches the expression on the right side exactly. Since both sides are identical for any value of \( x \), the equation is always true. This is known as an identity equation. It's useful to recognize identity equations because they indicate inherent truths about mathematical relationships. When each term on one side of the equals sign has an exact counterpart on the other side, we encounter this type of equation, making it true universally.

Prealgebra

Prealgebra is the foundational level of math that introduces basic algebraic concepts. It bridges the gap between arithmetic and algebra, preparing students for more advanced courses. In prealgebra, you'll encounter equations and concepts like variables, simple inequalities, and basic operations with fractions and decimals. Key concepts in prealgebra include:

• Understanding and working with variables.
• Simplifying expressions.
• Solving simple equations, identifying scenarios with no solution, and recognizing identity equations.
• Basic operations and properties of numbers.

Mastering prealgebra helps build confidence in handling algebraic equations and sets the stage for success in future math courses. It's important for recognizing different equation types such as those with no solution or those that are identities, as these concepts provide a deeper understanding of how equations behave.