Problem 38 - Elementary Algebra

Question

\(3 z+5 z=z(3+5)\)

Step-by-Step Solution

Answer

Both sides simplify to \(8z\).

Step 1: Simplify Left-Hand Side

Combine the like terms on the left-hand side: \(3z + 5z\).

Step 2: Compute Summation

Add the coefficients of \(z\): \(3 + 5 = 8\). This gives us \(8z\).

Step 3: Simplify Right-Hand Side

Multiply the term inside the parenthesis by \(z\): \(z(3 + 5)\).

Step 4: Compute Multiplication

Simplify the expression inside the parenthesis first: \(3 + 5 = 8\). This gives us \(z \times 8\), which simplifies to \(8z\).

Step 5: Verify Equality

Check if the simplified forms of the left-hand side and right-hand side are equal: \(8z = 8z\).

Key Concepts

Simplifying Expressions

When you simplify an algebraic expression, your goal is to make it as straightforward as possible. This often involves combining like terms and performing basic arithmetic operations.
In the given exercise, the expression is simplified as follows:

• First, identify the like terms. Here, both terms on the left-hand side of the equation, 3z and 5z, contain the variable z.
• To combine these terms, you simply add their coefficients (the numbers in front of the variable). So, 3z + 5z becomes 8z since 3 + 5 equals 8.
• You then write down the simplified form, which is 8z.

Simplifying expressions is about making them easier to understand and work with, which is a crucial skill in algebra. After simplification, the expression becomes much simpler and paves the way for further steps in problem-solving.

Algebraic Addition

Algebraic addition involves combining like terms to simplify an expression. Terms are considered 'like' if they have the same variable part.
Let's break this down with our exercise:

• First, we have two like terms: 3z and 5z.
• To add these, keep the variable part (z) unchanged, and only add the numerical coefficients: 3 + 5.
• This results in 8. So, 3z + 5z becomes 8z.

Remember, only like terms can be added together. If we had a term like 3z and another term like 5y, we could not combine them through addition because they contain different variables. Adding terms correctly is essential for solving algebraic equations and ensuring accuracy in mathematical solutions.

Verifying Equality

Verifying equality means checking if two expressions are indeed equal after simplification.
Let's take our simplified terms from the exercise:

• Step 1: Simplify both sides of the equation. In our example, both the left-hand side (3z + 5z) and right-hand side (z(3 + 5)) simplified to 8z.
• Step 2: Compare the simplified forms. Here, 8z on the left-hand side matches 8z on the right-hand side.
• Step 3: Conclude that the expressions are equal since both sides are identical.

Verifying equality ensures that our simplifications and calculations are correct. This step is crucial to validating the solutions to algebraic problems, ensuring the integrity of our work and preventing errors.