Problem 23 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{1}{x}=\frac{4}{3 x}+1 $$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{1}{3} \).

Step 1: Clear the Denominator

To eliminate the fractions, multiply every term in the equation by the common denominator, which is \( 3x \). This gives us:\[3x \cdot \frac{1}{x} = 3x \cdot \frac{4}{3x} + 3x \cdot 1\]Simplifying, this becomes:\[3 = 4 + 3x\]

Step 2: Rearrange the Equation

Subtract 4 from both sides of the equation to isolate the term with \( x \):\[3 - 4 = 3x\]This simplifies to:\[-1 = 3x\]

Step 3: Solve for x

Divide both sides of the equation by 3 to solve for \( x \):\[x = \frac{-1}{3}\]

Key Concepts

Fractions

Fractions are a way of representing parts of a whole or a division of quantities. In the given exercise, fractions appear prominently in the initial equation \( \frac{1}{x} = \frac{4}{3x} + 1 \). Each term is either a fraction or can be converted to one when dealing with equations.
Understanding how to manipulate these is critical when working with equations. To simplify fractions, we often look for common denominators. This makes operations like addition and subtraction easier because having like terms allows us to work with the numerators directly. In the exercise, multiplying through by the common denominator of \( 3x \) helps us clear the fractions entirely.

• When clearing fractions, choose a common multiple of all denominators involved.
• Fractions are often involved in division problems, appearing in rational expressions.
• Always aim to simplify fractions when possible; it often makes other algebraic operations more manageable.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the given equation, algebraic techniques are used to solve for \( x \). Let's explore some fundamental aspects:Algebra often involves rearranging equations to isolate a variable. Here, after clearing fractions, the goal is to get \( x \) by itself on one side of the equation.
This involves moving numbers across the equation using basic operations such as addition, subtraction, multiplication, and division.

• Identifying like terms helps simplify and clarify an equation.
• Algebra makes heavy use of inverse operations (e.g., subtracting to cancel addition).
• Rearranging terms can help see the solution better by isolating the variable.

For the given exercise, after clearing the denominators, the equation \( 3 = 4 + 3x \) requires subtraction to bring the equation to \( 3x = -1 \) before dividing both sides by 3 to solve for \( x \). Understanding these steps highlights the importance of basic algebraic manipulations in problem-solving.

Solving Equations

Solving equations is about finding the value of the variable that makes the equation true. The linear equation here is ultimately a simple form.In essence, this involves ensuring both sides of the equation are equal by finding the correct value of \( x \).In this problem, solving begins with the removal of fractions by multiplying through by the common denominator. Once fractions are cleared, the focus shifts to isolating the variable.

• Start by simplifying both sides if possible.
• Use operations that "undo" each other, such as addition versus subtraction or multiplication versus division.
• Constant terms (like \( 4 \) in this example) are often moved across the equation to facilitate solving.

After isolating \( 3x \), dividing both sides by 3 yields the solution \( x = \frac{-1}{3} \). The ability to reverse operations confidently helps in achieving the solution efficiently.
Equations like this one form the backbone of algebraic problem-solving and are foundational for more complex equations as well.