Problem 51 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given values of the variables. (Lesson 1.2 ) \((x+y)^{2}\) when \(x=5\) and \(y=2\)

Step-by-Step Solution

Answer

The evaluated expression for \(x = 5\) and \(y = 2\) is \(49\)

Step 1: Substitute the Given Values

Substitute the given values, \(x = 5\) and \(y = 2\) into the expression \((x + y)^{2}\) to get \((5+2)^{2}\)

Step 2: Simplify Inside Parentheses

According to the order of operations, what is inside the parentheses should be computed first. So, \(5+2\) equals \(7\). This simplifies the expression to \(7^2\).

Step 3: Evaluate the Resulting Expression

The final step is to evaluate \(7^2\), which equals \(49\)

Key Concepts

Substituting Values

Substituting values in algebraic expressions is like finding the missing piece of a puzzle. It's where the abstract meets the concrete. Let’s imagine that an expression is a recipe, and the variables are the ingredients. Substituting values is akin to replacing the ingredients with actual amounts, transforming general instructions into a specific formula for your delicious dish.

For instance, in the expression \( (x+y)^2 \) when we're given \( x=5 \) and \( y=2 \), we are simply plugging in these numbers in place of \( x \) and \( y \). Think of it as replacing placeholders with real numbers. In our recipe, it’s the moment you know that 'sugar' means '5 cups' and 'flour' means '2 cups'. The process converts the expression into \( (5+2)^2 \) which is much easier to work with and gets us one step closer to the final answer.

Order of Operations

Remember the phrase 'Please Excuse My Dear Aunt Sally'? This isn't just a quirky statement; it’s a mnemonic for the order of operations in mathematics: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Following this sequence is crucial to evaluate expressions correctly.

When dealing with the expression \( (x+y)^2 \) after substituting \( x=5 \) and \( y=2 \), we first focus on the operation inside the parentheses. This is our 'P' for parentheses. We add the numbers together to get \( 7 \). Next, we raise this sum to the power of 2, which is our 'E' for exponents. Skipping or rearranging steps will lead to an incorrect result, just like how adding ingredients in the wrong order can spoil the dish. Therefore, the correct order of our simplified expression is to calculate \( 7^2 \) only after we’ve figured out \( 5+2 \) first.

Simplifying Expressions

Simplifying expressions is like tidying up a room so it's easier to navigate. It's all about making an expression as straightforward as possible, without changing its value. In our given problem, once we've substituted values and adhered to the order of operations, we simplify the expression by performing the exponentiation.

After substituting and adding, we reached \( 7^2 \). To simplify \( 7^2 \) we multiply 7 by itself, which gives us 49. Thus, the expression \( (x+y)^2 \) when \( x=5 \) and \( y=2 \) simplifies to 49. This final answer is neat and tells us precisely what the original expression is equivalent to with the given values. Simplification is the final stroke of the brush that completes the masterpiece—the clean, clear, and final result of our algebraic expression.