Problem 33 - Elementary and Intermediate Algebra

Question

Solve each equation. If an equation is an identity or a contradiction, so indicate. $$ 2 y+1=5(0.2 y+1)-(4-y) $$

Step-by-Step Solution

Answer

The equation is an identity: \(2y + 1 = 2y + 1\).

Step 1: Distribute the Right Side

First, we need to distribute the terms on the right side of the equation: \(5(0.2y + 1) - (4-y)\).Distribute \(5\) to both \(0.2y\) and \(1\):\[5 imes 0.2y + 5 imes 1 = y + 5\]Now distribute the negative sign across \((4-y)\):\[-(4-y) = -4 + y\]So the expression becomes:\[y + 5 - 4 + y\] which simplifies to \(2y + 1\).

Step 2: Compare Both Sides of the Equation

The simplified right side expression from step 1 is \(2y + 1\).Now, let's write down the full equation with both sides simplified:\[2y + 1 = 2y + 1\]Both sides of the equation are identical, showing that they are the same for all values of \(y\).

Step 3: Identify the Type of the Equation

Since \(2y + 1 = 2y + 1\) holds true for all values of \(y\), this equation is an identity.An identity is an equation that is true for every possible value of the variable.

Key Concepts

Identities

In the world of algebra, an **identity** is a type of equation that is universally true for all valid choices of the variable involved. These equations are not dependent on specific values; no matter what value you substitute for the variable, the equation remains true. In our problem:
\[2y + 1 = 2y + 1\]We've discovered the equation is an identity in step 3 of the solution, as both sides of the equation are identical. Here are some tips to identify an identity:

• After simplifying both sides, see if they're exactly the same.
• Try substituting various numbers for the variable. If the equation holds for any number you try, you're likely dealing with an identity.

Understanding identities is crucial because they point to equations that provide no new information about the variable. They emphasize the importance of thorough equation simplification to reveal the nature of the problem at hand.

Equation Simplification

Simplifying an equation is like peeling away layers of complexity to reveal a more understandable core. The goal is to make equations easier to work with. It involves consolidating like terms, reducing expressions, and canceling terms if possible. Let's break down the steps involved in simplification:
- **Combine Like Terms:** Terms that have the same variable raised to the same power can be combined by summing or subtracting their coefficients.
- **Cancel Out Opposites:** If you have a term on both sides of the equation, like \(2y\) in our example, recognize that they cancel each other out when both sides are equal.

• In equations like \(2y + 1 = 2y + 1\), after simplification, you're often left with an identity.
• Simplification is crucial for recognizing whether an equation is simple, complex, or even a type like an identity or contradiction.

Understanding these steps is fundamental as it not only helps solve equations but also aids in spotting whether an equation is unique or belongs to a special class like identities.

Distributive Property

The **distributive property** is a key concept in algebra that allows you to multiply a single term by each term within a set of parentheses. It simplifies expressions and equations, helping us move forward in solving them. In the original exercise, the distributive property was used on the right side:
Using the distributive property:- Multiply \(5\) by each term inside the parentheses: \(5(0.2y + 1) = y + 5\).- Handle subtraction carefully: Remember that subtracting includes distributing the negative sign as well.In our equation:
\[5(0.2y + 1) - (4-y) = y + 5 - 4 + y\]Each step of the distribution allows terms to be properly expanded and recombined. This gives clarity and sets the stage for simplifying further. The distributive property is fundamental:

• For breaking down expressions into manageable parts.
• To ensure equations are broken down correctly, avoiding errors in signs and terms.

Recognizing and applying the distributive property effectively streamlines the solving process and is pivotal in mastering algebraic equations.