Problem 59 - Algebra A Combined Function

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5 .\) $$ 5 y^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 45.

Step 1: Substitution

Start by substituting the given values of variables into the expression. Here, you substitute \(y = 3\) into the expression \(5y^2\).

Step 2: Exponentiation

Calculate \(y^2\) by squaring the substituted value of \(y\). That is, compute \(3^2\), which equals \(9\).

Step 3: Multiplication

Multiply the result of the exponentiation by 5. So, compute \(5 \times 9\), which equals 45.

Key Concepts

Substitution Method

When evaluating algebraic expressions, it's essential to substitute the specific values given for each variable into the expression. This method allows us to transform abstract mathematical symbols into concrete numbers we can work with. In our original exercise, three different variables are provided: \(x = 1\), \(y = 3\), and \(z = 5\). Since the expression to be evaluated involves \(y\), you only need to substitute the value of \(y\) into it. So, replace every instance of \(y\) within the expression \(5y^2\) with 3. Doing this substitution is just like filling in the blanks with given numbers in each part of the equation.
Here's a simple reminder for substitution steps:

• Identify the variables in the expression.
• Look for their given values.
• Replace the variable in the expression with its specific value.

Substitution reduces the complexity by allowing us to work with pure numbers, facilitating subsequent calculations.

Exponentiation

Once the substitution is complete, the next step is exponentiation. Exponentiation involves raising a number to a power. In this specific exercise, after replacing \(y\) with 3, you need to compute \(y^2\) or \(3^2\). Exponentiation like \(3^2\) means multiplying the base, which is 3, by itself: 3 times 3. Explaining it step-by-step:

• Take the base: here, it's 3.
• Multiply the base by itself as many times as indicated by the exponent (2 times in this case).
• Thus, \(3 \times 3 = 9\).

Exponentiation is crucial for solving complex expressions since it allows us to simplify large repeated multiplications into more manageable numbers for further calculations.

Multiplication

After you've evaluated the expression inside the exponent, you reach the multiplication step. Here, you multiply the result of the exponentiation by any other coefficients outside. In this task, you have the number 5 multiplying \(3^2\) which we previously computed as 9. Multiplication is straightforward:

• Take the result from exponentiation, which is 9.
• Multiply it by 5: \(5 \times 9\).
• The product is 45.

Multiplication helps in combining the calculated squared value with the original coefficient from the expression, completing the operation.
This step emphasizes the importance of knowing basic multiplication rules, which are essential building blocks of algebra.

Evaluate Expressions

Evaluating expressions is the culmination of all previous steps: substitution, exponentiation, and multiplication. Each step simplifies and transforms the expression until only a single number remains. Here's how the process works in the example:

• Substitute \(y = 3\) into \(5y^2\).
• Exponentiate to find \(3^2 = 9\).
• Multiply 9 by 5 to get 45.

This sequential breaking down of the problem allows you to systematically tackle any algebraic expression. The goal is to replace variables with numbers and simplify the expression thoroughly until it's reduced to a clear and concise answer. Such practices in evaluating expressions form the foundation of solving more complicated mathematical problems.