Problem 91 - Algebra and Trigonometry

Question

Solve each 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 \(x = 124\).

Step 1: Simplify the equation

Start by performing the operations inside the brackets. Calculate \(6 \div 3\) and \(5^{3}\) first, then add the results, multiply by \(6 \div 3\) and finally subtract \(5x\). This gives us a new equation: \(5-12x = 8-7x - [(2+125)+5x]\). Further simplification results in \(5-12x = 8-7x -127 -5x\)

Step 2: Rearrange the equation to one side

Next, combine the like terms in the equation to get \(5+127 = 8 +x\). This simplifies to \( x = 132-8\).

Step 3: Solve for x

Finally, subtract 8 from 132 to solve for x, obtaining \( x = 124\)

Key Concepts

Equation Solving

When solving algebraic equations, the goal is to determine the value of the variable that makes the equation true. In our example, the equation is a complex expression:

• First, it is essential to simplify each side of the equation. Begin by solving any operations inside parentheses or brackets, as they take precedence.
• For instance, perform calculations such as division and powers within brackets.

Once simplified, ensure that all like terms are gathered on the same side of the equation. This process helps to balance the equation and make it easier to solve.
Rearranging terms and performing basic arithmetic operations will further simplify the process to arrive at the solution.
Equation solving requires logical thinking and practice, especially with more complex equations. Start by simplifying each part, move terms around to their proper place, and methodically work towards the solution.

Simplifying Expressions

Simplifying expressions is a crucial step in solving equations, as it helps to reduce complexity and reveals the core structure. In the provided equation, simplification involves handling operations within the brackets first.

• For example, calculate mathematical operations like division and exponents early on. In this case, calculating \(6 \div 3\) and \(5^3\) is the first step.
• Subsequently, add and subtract within the brackets, before proceeding to combine these results with any surrounding terms.

By simplifying each segment in a structured manner, you can prevent errors and make the remaining steps much simpler.
Each simplified expression allows for easier manipulation when isolating the variable later on.
Remember that simplification may involve combining like terms or reducing any coefficients, wherever possible.

Variable Isolation

Variable isolation is about rearranging the equation so that the variable appears alone on one side of the equals sign.
The principle technique involves moving terms that don't contain the variable to the opposite side of the equation:

• In our practice equation, after simplifying the expressions, move constants away from the variable.
• When like terms share both sides, subtract or add them as needed to help the variable stand alone.

Once isolated, you can solve for the variable directly. In the given example, isolating \(x\) helps us subtract 8 to find its value.
Variable isolation transforms a complex equation into a simple arithmetic operation that gives the value of the variable, thus solving the equation.
Progressive isolation through simplification and thoughtful manipulation is key to clear and accurate solutions.