Problem 91 - College Algebra Essentials

Question

Solve equation. \(5-12 x=8-7 x-\left[6 \div 3\left(2+5^{3}\right)+5 x\right]\)

Step-by-Step Solution

Answer

The solution to the equation is \( x = 25.7 \)

Step 1: Simplify the equation.

Begin by simplifying the expression inside the bracket: \( \left[6 \div 3\left(2+5^{3}\right)+5 x\right] \). The first operation to do is the exponent: \( 5^{3} = 125 \), and then the addition inside the parenthesis: \( 2 + 125 = 127 \). After that, the division: \( 6 \div 3 = 2 \), and finally multiplication: \( 2 \times 127 = 254 \). Thus, the original equation can be rewritten as: \( 5 - 12x = 8 - 7x - (254 + 5x) \).

Step 2: Gather like terms and remove the brackets.

Next, eliminate the brackets, and gather like terms by adding or subtracting them from both sides of the equation to get: \( 12x - 7x + 5x = 8 - 5 + 254 \).

Step 3: Simplify Again.

This equation simplifies to: \( 10x = 257 \).

Step 4: Solve for \( x \).

Finally, to solve for \( x \), you divide by the coefficient of \( x \), which is 10. Thus, \( x = \frac{257}{10} \) or \( x = 25.7 \)

Key Concepts

Simplifying Algebraic Expressions

When dealing with algebraic expressions, simplification is key to making the math more manageable. Simplification involves combining like terms, performing arithmetic operations, and reducing the expression to its simplest form. In the exercise, simplifying the expression within the brackets was the initial step. We started with the exponent calculation, followed by the operations of addition and division, resulting in a more concise expression. Simplification helps in reducing complexity, making it easier to manipulate the equation further.

Exponents

Exponents are a fundamental part of algebra, representing repeated multiplication of a number by itself. For example, in the exercise, we dealt with an exponent in the term \(5^3\), which means multiplying 5 by itself three times: \[ 5^3 = 5 \times 5 \times 5 = 125 \]Always remember to solve exponents as part of the order of operations. Exponents can significantly change the value of terms, so they should be resolved early when simplifying an algebraic expression. Understanding exponents can simplify the problem-solving process, allowing you to work with more straightforward terms.

Equations with Variables

Equations with variables require us to find the value of the unknown that makes the equation true. In our exercise, the variable is \( x \), and the goal is to solve for \( x \). Variables are placeholders and can be subject to various arithmetic operations. Learning to isolate the variable by performing operations like addition, subtraction, multiplication, or division is crucial. This is often done by manipulating the equation to form simpler expressions, eventually leading to a straightforward solution for the variable.

Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)), is essential in solving algebraic equations correctly. In the given exercise, this order guided the simplification process. Firstly, operations within parentheses were addressed—even after simplifying the terms inside brackets with operations like addition and exponents. Then, division and multiplication were completed in sequence. Following this order ensures accuracy and consistency in solving equations, eliminating potential errors in computation. Practicing this systematic approach will enhance problem-solving efficiency in algebra.