Problem 41 - Algebra 1

Question

Evaluate the expression for the given values of the variables. $$ (x+y)^{2} \text { when } x=5 \text { and } y=3 $$

Step-by-Step Solution

Answer

The value of the expression \((x+y)^{2}\) when \(x = 5\) and \(y = 3\) is 64.

Step 1: Substitution

Replace the \(x\) and \(y\) in the equation \((x+y)^{2}\) with the given values. This will give: \((5+3)^{2}\).

Step 2: Perform the Addition Operation

Perform the addition operation inside the parentheses before squaring. The equation becomes: \(8^2\).

Step 3: Perform the Squaring Operation

Now square the result from the previous step: \(8^2 = 64\).

Key Concepts

Substitution in Algebra

Substitution in algebra is a fundamental concept that involves replacing variables in an expression or equation with numerical values. This process makes it easier to evaluate the expression or solve the equation. Here’s a simple way to think about it: each variable in your expression is like a placeholder, and substitution is simply the act of filling in those placeholders with the numbers specified.

• Identify the variables in your expression.
• Substitute the values given for those variables.
• Solve the resulting numerical expression.

For the exercise given, the expression \((x + y)^2\) is simplified by substituting the values provided: \(x = 5\) and \(y = 3\). After substitution, the expression transforms into \((5 + 3)^2\). This clarity from substitution helps us focus on simplifying and evaluating the numerical expression. It’s a step that sets the stage for tackling more complex expressions or equations.

Order of Operations

The order of operations is crucial in correctly evaluating mathematical expressions. Known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), this rule outlines the sequence you should follow when working through an expression.
In the exercise example \((5+3)^2\), following the order of operations means starting with tasks inside parentheses.

• First, compute \(5+3\), which gives \(8\).
• Next, move to the exponent, \(8^2\), which results in \(64\).

The order of operations ensures consistency and accuracy. Without it, solving the same expression in different ways might lead to different results, which in a mathematical sense, is not correct. So, always remember to follow PEMDAS!

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations (such as addition and multiplication). They can be simple, made up of just one term, or complex, with multiple terms and operations.
In our exercise, the expression \((x+y)^2\) is a polynomial expression involving addition and exponentiation. An expression like this can represent numerical scenarios in real life or serve as a mathematical model for more extensive calculations.

• Variables: they can stand for unknown, placeholders for numbers.
• Operations: determine how the numbers and variables are combined or manipulated.

Understanding algebraic expressions is essential because it forms the foundation for algebraic manipulation and solving equations. Each part of an expression corresponds to a different piece of the mathematical puzzle you're aiming to solve. Mastering how to read and simplify these expressions can significantly ease problem-solving tasks in algebra.