Solve for x (5x-2)/(3x)=7

Solution:

Step 1: Determine the Least Common Denominator (LCD)

Identify the LCD of all the fractions involved in the equation.

Step 1.1: LCD of Values

The LCD for the values $3x$ and $1$ is equivalent to finding the Least Common Multiple (LCM) of their denominators.

Step 1.2: LCM of Denominators

The LCM of $1$ and any other number or expression is simply that number or expression itself, which in this case is $3x$.

Step 2: Clear the Fractions

Eliminate the fractions by multiplying every term in the equation $\frac{5x - 2}{3x} = 7$ by $3x$.

Step 2.1: Multiply Each Term

Multiply both sides of the equation $\frac{5x - 2}{3x} = 7$ by $3x$ to clear the fractions.

Step 2.2: Simplify the Left Side

Step 2.2.1: Use Commutative Property

Apply the commutative property of multiplication to rearrange the terms: $3x \cdot \frac{5x - 2}{3x} = 7 \cdot 3x$.

Step 2.2.2: Cancel Common Factors

Step 2.2.2.1: Factor Out $3$

Factor out the $3$ from $3x$ in the fraction.

Step 2.2.2.2: Perform Cancellation

Cancel out the common factor of $3$ from the numerator and denominator.

Step 2.2.2.3: Rewrite the Expression

After cancellation, the expression becomes $\frac{5x - 2}{x} \cdot x = 7 \cdot 3x$.

Step 2.2.3: Cancel Common Factor of $x$

Step 2.2.3.1: Perform Cancellation

Cancel out the common factor of $x$ from the numerator and denominator.

Step 2.2.3.2: Rewrite the Expression

The simplified expression is now $5x - 2 = 7 \cdot 3x$.

Step 2.3: Simplify the Right Side

Step 2.3.1: Multiply $7$ by $3x$

Multiply $7$ by $3x$ to get $21x$: $5x - 2 = 21x$.

Step 3: Solve for $x$

Step 3.1: Move $x$ Terms to One Side

Get all terms containing $x$ on one side of the equation.

Step 3.1.1: Subtract $21x$ from Both Sides

Subtract $21x$ from both sides: $5x - 2 - 21x = 0$.

Step 3.1.2: Combine Like Terms

Combine the $x$ terms: $-16x - 2 = 0$.

Step 3.2: Isolate the Constant Term

Add $2$ to both sides to isolate the $x$ term: $-16x = 2$.

Step 3.3: Divide to Solve for $x$

Step 3.3.1: Divide Each Side by $-16$

Divide both sides of the equation by $-16$: $\frac{-16x}{-16} = \frac{2}{-16}$.

Step 3.3.2: Simplify the Left Side

Step 3.3.2.1: Cancel Common Factors

Cancel the common factor of $-16$.

Step 3.3.2.1.1: Perform Cancellation

After cancellation, the left side becomes $x$.

Step 3.3.2.1.2: Divide $x$ by $1$

The equation simplifies to $x = \frac{2}{-16}$.

Step 3.3.3: Simplify the Right Side

Step 3.3.3.1: Cancel Common Factors

Step 3.3.3.1.1: Factor Out Common Factors

Factor out the common factors of $2$ and $-16$.

Step 3.3.3.1.2: Perform Cancellation

After canceling the common factor, the fraction reduces to $x = \frac{1}{-8}$.

Step 3.3.3.2: Position the Negative Sign

Place the negative sign in front of the fraction: $x = -\frac{1}{8}$.

Step 4: Express the Result in Different Forms

The solution can be expressed as:

Exact Form: $x = -\frac{1}{8}$ Decimal Form: $x = -0.125$

Knowledge Notes:

The problem-solving process involves several key mathematical concepts:

1. Least Common Denominator (LCD): The smallest common multiple of the denominators of two or more fractions. It is used to combine fractions into a single fraction or to eliminate fractions from an equation.

2. Least Common Multiple (LCM): The smallest multiple that is exactly divisible by every number of a set. When dealing with fractions, the LCM of the denominators is used as the LCD.

3. Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not change the product. For example, $ab = ba$.

4. Cancellation: A method used to simplify expressions by dividing both the numerator and the denominator by a common factor.

5. Combining Like Terms: A process used to simplify an expression or equation by adding or subtracting terms that have the same variables raised to the same power.

6. Isolating the Variable: The process of manipulating an equation to get the variable of interest, often represented as $x$, by itself on one side of the equation.

7. Simplifying Expressions: Reducing expressions to their simplest form by performing arithmetic operations and combining like terms.

Understanding these concepts is crucial for solving algebraic equations and simplifying mathematical expressions.