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

The problem presents an equation in which you are asked to solve for the variable x. The equation involves a fraction with the expression (5x-2) in the numerator and the variable x in the denominator, multiplied by the scalar 3. This entire expression is then set equal to the number 7, and the task is to manipulate the equation using algebraic methods to find the value of x that satisfies this equation.

$\frac{5 x - 2}{3 x} = 7$

Answer

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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 $3 x$ 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 $3 x$ .

Step 2: Clear the Fractions

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

Step 2.1: Multiply Each Term

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

Step 2.2.2: Cancel Common Factors

Step 2.2.2.1: Factor Out $3$

Factor out the $3$ from $3 x$ 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 $\frac{5 x - 2}{x} \cdot x = 7 \cdot 3 x$ .

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 $5 x - 2 = 7 \cdot 3 x$ .

Step 2.3: Simplify the Right Side

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

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

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 $21 x$ from Both Sides

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

Step 3.1.2: Combine Like Terms

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

Step 3.2: Isolate the Constant Term

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

Step 3.3: Divide to Solve for $x$

Step 3.3.1: Divide Each Side by $- 16$

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

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:

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.
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.
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not change the product. For example, $a b = b a$ .
Cancellation: A method used to simplify expressions by dividing both the numerator and the denominator by a common factor.
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.
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.
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.