Problem 3 - Algebra 2

Question

Evaluate each expression if \(x=4, y=-2,\) and \(z=3.5\) \(x+[3(y+z)-y]\)

Step-by-Step Solution

Answer

The evaluated expression is 10.5.

Step 1: Substitute Given Values

First, substitute the given values into the expression. Replace \(x\) with 4, \(y\) with -2, and \(z\) with 3.5, resulting in:\[4 + [3(-2 + 3.5) - (-2)]\]

Step 2: Simplify the Parentheses

Calculate the expression within the parentheses. Start with the inner parentheses:\(-2 + 3.5 = 1.5\)Now substitute back:\[4 + [3(1.5) - (-2)]\]

Step 3: Multiply Inside the Brackets

Continue by multiplying inside the brackets:\[3 \cdot 1.5 = 4.5\]Now substitute this back into the expression:\[4 + [4.5 - (-2)]\]

Step 4: Simplify the Expression Inside the Brackets

Simplify the expression inside the brackets:\[4.5 - (-2) = 4.5 + 2 = 6.5\]Now the expression becomes:\[4 + 6.5\]

Step 5: Final Addition

Add the final terms outside the brackets:\[4 + 6.5 = 10.5\]

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that involves replacing variables in an expression with their corresponding given values. This helps in evaluating the expression to yield a concrete numerical result. In the given exercise, we are tasked with evaluating the expression \(x + [3(y+z)-y]\) given the values of \(x=4\), \(y=-2\), and \(z=3.5\). To perform substitution, we simply replace:

• \(x\) with \(4\)
• \(y\) with \(-2\)
• \(z\) with \(3.5\)

This results in the substituted expression: \(4 + [3(-2 + 3.5) - (-2)]\). Substitution simplifies the algebraic expression to a mathematical problem focusing on numbers, making it more straightforward to solve.

Simplification

Simplification is the process of reducing an expression to its simplest form. This involves performing operations inside any parentheses first and combining like terms in the expression. In our given problem, simplification starts after substitution. We first calculate the inner parentheses portion, \((-2 + 3.5)\). By adding these numbers, we get \(1.5\). So the expression now becomes \(4 + [3(1.5) - (-2)]\). Next, we perform operations inside the square brackets to further reduce the complexity. We simplify the terms by multiplication and subtraction, until we can't reduce it further. This step prepares us for the final evaluation of the expression.

Arithmetic Operations

Arithmetic operations form the basis of simplifying and evaluating expressions. These include addition, subtraction, multiplication, and division. Within our exercise, each of these operations plays a role. After finding \(1.5\) from the parentheses, we multiply it by \(3\), obtaining \(4.5\). Following this, the expression turns into \(4 + [4.5 - (-2)]\). Here, subtraction and understanding negative numbers come into play. The expression \(4.5 - (-2)\) is equivalent to \(4.5 + 2\), which gives us \(6.5\). By applying arithmetic operations, we methodically approach the simplest form of the expression, ensuring no computational step is overlooked.

Order of Operations

The order of operations is a key principle to correctly evaluating expressions. It serves as a rulebook guiding which operations to perform first, ensuring consistency in solutions. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), this order ensures accurate results. In our problem, we start by evaluating any expressions inside parentheses. Operations inside the square brackets are completed next, following multiplication and correct handling of subtraction signs. After simplifying inside the brackets, we proceed with a straightforward addition of \(4 + 6.5\), resulting in \(10.5\). Sticking to this order prevents errors and leads us methodically through the problem.