Problem 47 - Beginning and Intermediate Algebra

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: Understand the Expression

The expression given is \(3y\). It means 3 times the value of \(y\).

Step 2: Substitute the Value of y

Replace \(y\) in the expression \(3y\) with the given value \(y=3\). The expression becomes \(3 \times 3\).

Step 3: Perform the Multiplication

Multiply the numbers in the expression: \(3 \times 3 = 9\). This is the value of the expression when \(y=3\).

Key Concepts

Substitution

In algebra, substitution is a crucial step when working with algebraic expressions. Substitution involves replacing variables in an expression with given numerical values. This allows you to transform an algebraic expression into a numerical one that can be solved easily.
For example, if you have a variable, like in the expression \(3y\), and you are given that \(y = 3\), you can substitute \(3\) for \(y\). The expression becomes \(3 \times 3\). This is handled just like a standard numeric operation. By substituting the variable, you simplify the expression and make it possible to find a solution. It's an essential method for evaluating expressions in homework or exercises.

Multiplication

Multiplication is one of the basic arithmetic operations, and it's essential for evaluating algebraic expressions. When you have an expression like \(3y\), this instructs you to multiply the constant 3 with the value of the variable \(y\).
Once substitution is complete, you perform multiplication to simplify the expression. In our example, after substituting \(y=3\), the expression becomes \(3 \times 3\).

• Multiplication involves finding the product of the numbers.
• It's commutative, meaning the order in which you multiply does not change the result. For instance, \(3 \times 3\) is the same as \(3 \times 3\).

Completing multiplication allows you to move from an algebraic statement to a numeric solution.

Variables

Variables are symbols used to represent unknown numbers or quantities in mathematics. Commonly, letters such as \(x\), \(y\), or \(z\) are used as variables. They make it possible to write general rules or formulas that apply to various specific cases.
In our example, \(y\) is a variable standing in for a number that can change based on the context of the problem. When given a specific value, like \(y = 3\), you substitute this number into the expression to evaluate it. Variables add a dynamic component to mathematics that allows for flexibility and generalization.

• They help create equations that model real-world situations.
• Variables are fundamental for solving equations and understanding functions.

Understanding how to manipulate and substitute variables is vital for progressing in algebra and beyond.