Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the variable. \(8(x+3)-64\) for \(x=2\)

Step-by-Step Solution

Answer

The solution to the expression is \(-24\).

Step 1: Simplify the Expression Inside the Parentheses

Start by substituting the given value of \(x\), which is 2, into the equation. This gives us: \[8((2)+3)-64\]

Step 2: Calculate the Value Inside the Parentheses

Perform the addition inside the parentheses. This turns the expression into: \[8(5)-64\]

Step 3: Multiply by the Coefficient

Next, multiply the result from inside the parentheses by the coefficient 8. This gives:\[40-64\]

Step 4: Subtract Constant from Product

Finally, subtract 64 from 40, resulting in the following:\[40-64 = -24\]

Key Concepts

Variable Substitution

When solving algebraic equations, the process often involves a technique known as **variable substitution**. This means replacing a variable in the equation with a given number. In the problem we have, the variable \(x\) is replaced with 2. This substitution allows us to turn an algebraic expression into a numerical one.

Here’s how it works in our example: The original expression is \(8(x+3)-64\). Once we substitute the \(x\) with 2, the expression looks like \(8(2+3)-64\). This makes the equation easier to solve because it contains only numbers.

Substitution is the foundational step for simplifying equations and determining their values. It transforms the problem into a series of simpler arithmetic operations, which ultimately leads to the solution.

Simplifying Expressions

Once you have substituted the variable with a number, the next step is **simplifying expressions**. This involves performing operations such as addition, subtraction, multiplication, and division as per the maths involved in the equation.

In our example, after substituting \(x=2\), we simplify the expression inside the parentheses first. This is crucial because it reduces complexity step by step. So, \((2+3)\) simplifies to 5, which changes our expression to \(8 \times 5 - 64\).

By systematically breaking down expressions into simpler terms, you can follow through each step with clarity, which reduces the chance of mistakes. Simplifying doesn't only make solving easier; it also improves understanding of how different parts of an equation interact.

Order of Operations

One of the most crucial aspects of solving an equation is the **order of operations**, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In the given problem, after substituting \(x=2\) and simplifying the expression inside the parentheses, the order of operations helps us decide the sequence of steps:

• First, resolve the value inside parentheses\((2+3)\)
• Next, multiply \(5\) by \(8\)
• Finally, perform subtraction \(40-64\)

Following PEMDAS ensures accuracy and that all mathematical rules are adhered to. It is the framework that keeps calculations consistent and reliable, which is essential for reaching the correct answer.