Problem 11 - Prealgebra

Question

Evaluate each expression if \(x=-2\) and \(y=4\) $$5(y-1)^{2}$$

Step-by-Step Solution

Answer

The expression evaluates to 45 when \(x=-2\) and \(y=4\).

Step 1: Substitute Values

First, substitute the given values for each variable in the expression. Replace \( y \) with 4.

Step 2: Simplify Inside the Parentheses

Calculate the expression inside the parentheses: \( y - 1 \), which becomes \( 4 - 1 = 3 \).

Step 3: Square the Result

Next, square the result from the previous step: \( 3^2 = 9 \).

Step 4: Multiply by Coefficient

Multiply the squared value by the coefficient 5: \( 5 \times 9 = 45 \).

Key Concepts

Understanding Substituting Values

Substituting values is a fundamental concept in algebra that involves replacing variables with their given numerical values. In this exercise, the expression given is \(5(y-1)^{2}\). Here, the variable \(y\) is provided as 4.

Imagine variables as placeholders that can take on different numbers. When you're given a specific value for a variable, like \(y = 4\), you simply replace the variable in the expression with that number. This is straightforward but crucial for accurately solving the expression.

• Replace \(y\) in the expression with 4

This step sets the stage for further simplification because it allows us to work with concrete numbers rather than abstract symbols.

Simplifying Expressions Step by Step

Simplifying expressions involves performing operations to make an expression as simple as possible. After substituting \(y\) in the expression \(5(y-1)^{2}\), we are left with \(5(4-1)^{2}\). The next step is to simplify what's inside the parentheses.

You have to work inside the parentheses first, just like solving any type of equation. For \(4 - 1\):

• Subtract 1 from 4 to get 3

Now the expression is \(5(3)^{2}\). Performing operations within parentheses first is crucial as per the order of operations, ensuring clarity and correctness as you simplify.

Understanding and Using Exponents

Exponents represent repeated multiplication of a base number. In our expression, once simplified inside the parentheses, we have \(3^2\). This means 3 squared, or 3 multiplied by itself once.

The definition of exponents makes it straightforward:

• \(3^2\) means \(3\times3\)
• Calculate to get 9

Exponents are fundamental because they allow compact expression of multiplication. Remember, squaring a number results in a larger value when multiplied by itself, contributing to an accurate solution for the expression.

Mastering the Order of Operations

The order of operations is a rule to evaluate mathematical expressions correctly. It ensures that calculations are performed in the right sequence to give the correct answer. The typical order follows the acronym PEMDAS:

- Parentheses
- Exponents
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)

In our problem of \(5(y-1)^{2}\), after simplifying to \(5(3)^{2}\), we first handled the parentheses, then the exponent, and finally, adjust for multiplication by 5.

• Multiply 9 by 5 to get the final answer, 45.

Understanding and applying this sequence is essential for solving expressions correctly and avoids mistakes that can arise from randomly doing calculations.