Problem 61 - Algebra A Combined Function

Question

Solve for \(x\). $$ 6 x-3(2-5 x)=6 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{4}{7}\).

Step 1: Expand the Expression

First, we need to expand the expression inside the parentheses by distributing the -3 in the term \[-3(2-5x)\]. This results in:\[-3 imes 2 - 3 imes (-5x) = -6 + 15x.\]

Step 2: Simplify the Equation

Substitute the expanded expression back into the original equation. This gives you:\[6x - 6 + 15x = 6.\]Combine like terms:\[21x - 6 = 6.\]

Step 3: Add 6 to Both Sides

To isolate the term with \(x\), add 6 to both sides of the equation. This results in:\[21x = 12.\]

Step 4: Solve for x

Divide both sides of the equation by 21 to solve for \(x\):\[x = \frac{12}{21}.\]Simplify the fraction:\[x = \frac{4}{7}.\]

Key Concepts

Solving Linear Equations

Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. Learning how to solve these equations is a fundamental skill in algebra that sets the foundation for tackling more complex problems. In the given exercise, we have the equation:
The goal is to find the value of the variable, typically denoted as \(x\). Solving for \(x\) involves a set of strategies and methods.

• Start by analyzing the structure of the equation. Identify terms containing the variable and constants.
• Apply operations such as addition, subtraction, multiplication, or division to both sides of the equation to maintain its balance.
• The ultimate aim is to isolate the variable on one side of the equation.

Mastering these steps enables you to handle a wide variety of algebraic problems successfully.

Step by Step Algebra

Algebra is often most effectively learned through a step-by-step approach. This process helps to break down the problem into manageable parts and ensures accuracy in the solution.
In this exercise, the problem is addressed methodically:

• Expanding Expressions: The initial step is to expand any expressions, which involves distributing any constants through parentheses.
• Combining Like Terms: Next, combine terms that have the same variable. This simplification is crucial to reduce the complexity of the equation.
• Isolating the Variable: Move all terms involving the variable to one side of the equation.
• Solving: Perform the necessary arithmetic operations to solve for the variable.

Each of these steps contribute to a thorough understanding and solving of algebraic equations. It's essential to practice these steps consistently for mastery.

Simplifying Expressions

Simplifying expressions is a valuable skill in algebra, as it makes equations easier and quicker to solve. It involves several techniques that work together to tidy up algebraic expressions:

• Distributing Coefficients: When dealing with parentheses, distribute any external number or sign to every term inside the bracket.
• Combining Like Terms: Any terms that share the same variable can be summed or subtracted, which simplifies the overall equation.
• Fraction Reduction: Fractions often appear in solutions, and reducing them to their simplest form is an important step.

In the example given, distributing and combining terms allowed the equation to be simplified effectively. This technique not only clarifies the problem but aids in solving it accurately. Simplification is the key to efficient problem-solving in algebra.