Problem 49 - Algebra A Combined Function

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5 .\) \(3 y\)

Step-by-Step Solution

Answer

The expression evaluates to 9.

Step 1: Substitute the given values into the expression

We start by substituting the given value of the variable \(y\) into the expression \(3y\). Since we are given that \(y = 3\), we replace \(y\) with 3 in the expression. This changes our expression to \(3 \times 3\).

Step 2: Multiply the numbers

Now we will multiply the numbers in the expression. Multiply 3 by 3, which is written as \(3 \times 3\). The result of this multiplication is 9. Thus, the evaluated expression for \(3y\) when \(y = 3\) is 9.

Key Concepts

Substitution Method

The substitution method is an essential strategy in solving algebraic expressions. It involves replacing variables with given values to simplify and evaluate expressions. For example, in this exercise, the expression to be evaluated is \(3y\). With a given value of \(y = 3\), substitution involves replacing \(y\) with 3 directly in the expression.
This step turns \(3y\) into \(3 \times 3\). It's a straightforward process yet crucial in correctly interpreting the problem statement. The accuracy of evaluating an expression largely depends on making the right substitutions. Remember:

• Identify the variable in the equation.
• Look for the given values in the problem.
• Replace the variable with its corresponding value.

Substitution provides a concrete way to work with algebraic expressions by simplifying them to a form that's easier to handle numerically.

Basic Algebra

Basic algebra is all about understanding how numbers and operations form expressions and how these expressions can be simplified. Algebra often involves finding the value of expressions for given variable values. In the expression \(3y\), understanding that \(y\) stands for a number is fundamental. By performing substitution, as shown earlier, the expression morphs into arithmetic that allows for direct calculations.
Algebra uses rules like:

• Operations precedence - multiplication before addition or subtraction if no brackets indicate otherwise.
• Balancing equations - maintaining equality when manipulating both sides of an equation.

Though this exercise is relatively simple, mastering these foundational ideas enables students to tackle more complex problems later. Bridging the gap from understanding letters and numbers to evaluating numerical results is the key role of basic algebra.

Multiplication

Multiplication is a core arithmetic operation needed for evaluating expressions like \(3y\). Once the substitution is performed, the expression \(3y\) becomes \(3 \times 3\), where \(3\) represents the coefficient and the substituted value of \(y\).
Understanding multiplication involves grasping that it is repeated addition. Here, \(3 \times 3\) can be perceived as adding 3 three times, resulting in 9. This operation is especially crucial in algebra where expressions are often based on coefficients and variable terms.
Key points to remember in multiplication:

• Multiplication is commutative: \(a \times b = b \times a\).
• Facilitates scaling: multiplying a variable or term scales its value.
• Essential for understanding advanced algebra like quadratic expressions or exponents.

Mastering multiplication not only aids in simpler problems but also prepares students for handling complex expressions efficiently.