Problem 77 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$\frac{6}{5} z+2 \text { when } z=5$$

Step-by-Step Solution

Answer

The value of the expression when \(z=5\) is \(8\)

Step 1: Substitute the Given Value

The variable \(z\) is given as 5. Substitute \(z=5\) into the expression. It becomes \(\frac{6}{5}(5)+2\)

Step 2: Simplify the Expression

First, simplify the multiplication in the fraction. The \(5\) in the numerator and the \(5\) in the denominator cancel each other out, leaving you with \(6+2\)

Step 3: Perform the Addition

Finally, add the two numbers to get \(8\)

Key Concepts

Substituting Variables

In algebra, substituting variables is a foundational skill that involves replacing letters or symbols with numbers or other expressions. The power of algebra comes from its use of variables--placeholders for unknown values that allow us to solve problems and create formulas. For instance, in the expression \( \frac{6}{5} z + 2 \), the letter \( z \) is a variable. To evaluate this expression for \( z = 5 \), we simply replace \( z \) with the number 5. The process might look like this: \( \frac{6}{5} \times 5 + 2 \).

Substituting correctly is essential because it sets the stage for solving the entire problem. If the substitution is done incorrectly, the rest of the solution will likely be incorrect as well. \(

• Always replace the variable with the given or known value.
• Make sure to maintain the structure of the original expression when substituting.

\)

Simplifying Expressions

Simplifying expressions is an important step following the substitution of variables. It's the process of making an algebraic expression more manageable, often by combining like terms and reducing fractions. In our example expression after the substitution, \( \frac{6}{5} \times 5 + 2 \), we notice that simplification is necessary. Here, the values 6 and 5 are in a fraction, but because we're multiplying by 5, we can cancel the numerator and the denominator since they are the same number. What's left is \( 6 + 2 \).

The goal of simplification is to present the expression in its most reduced form. In most cases, this involves:

• Combining like terms (terms with the same variable to the same power).
• Applying arithmetic operations such as addition, subtraction, multiplication, and division.
• Reducing fractions to their lowest terms.

Keep each of these steps in mind to accurately simplify expressions in algebra.

Arithmetic Operations in Algebra

Arithmetic operations in algebra aren't too different from the arithmetic you're used to, but they're carried out with variables as well as numbers. Once you have substituted variables and simplified expressions, you're often left with an expression that requires basic arithmetic to solve. For the given expression \( 6 + 2 \), we perform a simple addition to find the value of 8.

Understanding how to correctly execute arithmetic operations with variables is crucial, as they form the backbone of most algebraic manipulations. To ensure accuracy:

• Follow the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).
• Keep track of negative and positive signs.
• Be mindful of special rules, like the distributive property, when dealing with variables and constants.

Mastering these skills is key to successfully evaluating algebraic expressions.