Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the given equation. $$ \frac{r}{3 r-1}=4 $$

Step-by-Step Solution

Answer

The short answer to solving the equation \(\frac{r}{3r-1} = 4\) is to first clear the fraction by multiplying both sides by (3r-1), then simplify the equation to get \(r = 12r - 4\). Next, isolate r by subtracting 12r from both sides, yielding \(-11r=-4\). Finally, divide both sides by -11 to get the solution \(r = \frac{4}{11}\).

Step 1: Clear the fraction.

To clear the fraction, we have to multiply both sides of the equation by the denominator (3r - 1). This will give us a linear equation without the fraction. \((3r - 1) \cdot \frac{r}{3r - 1} = 4 \cdot (3r - 1)\)

Step 2: Simplify the equation.

Now, we can simplify the left side of the equation by cancelling out the (3r - 1) terms, and expand the right side of the equation using the distributive property. \(r = 12r - 4\)

Step 3: Solve for r.

In order to isolate r, we need to move all the terms with r on one side of the equation. Let's subtract 12r from both sides of the equation. \(r - 12r = - 4\) This simplifies to: \(-11r = - 4\) Now, we'll divide both sides by -11 to isolate r. \(\frac{-11r}{-11} = \frac{-4}{-11}\) This gives us the final solution: \(r = \frac{4}{11}\)

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. They are fundamental in algebra and appear in the form \(ax + b = c\). To solve linear equations, we aim to isolate the variable on one side of the equation, resulting in a simple expression. This involves the application of inverse operations to manipulate the equation without altering its equality.

• Start by getting rid of fractions, if any.
• Use addition or subtraction to move terms with the variable to one side.
• Use multiplication or division to isolate the variable.

Understanding linear equations thoroughly is crucial for mastering algebra. They represent constant rates of change and can be found in various real-world contexts. Being comfortable with these equations builds a strong foundation for more complex mathematical topics.

Fractional Equations

Fractional equations contain fractions with variables in the numerator, denominator, or both. Solving these can seem more complex but can be simplified effectively. The key is to get rid of the fraction by applying a clear method, often multiplying through by the least common denominator (LCD) of all fractions involved.
In our specific exercise, the equation \(\frac{r}{3r-1}=4\) involves a fraction that can be handled by multiplying every term by the denominator \(3r - 1\). This step clears the fraction and transforms the equation into a more straightforward linear equation.

• Multiply each term by the LCD to eliminate fractions.
• Simplify and rearrange the equation as needed.

This technique makes fractional equations much easier to solve and avoids the complexities often associated with fractions. It's a helpful trick to streamline your algebraic processes.

Equation Simplification

Simplifying equations is a crucial step when solving algebraic problems. This means reducing an equation to its simplest form, making it easier to solve. In the context of the given exercise, simplifying involves canceling terms and collecting like terms.

• Identify and cancel out common factors on both sides of the equation.
• Combine like terms to reduce complexity.
• Ensure all terms are as simplified as possible without losing equality.

By simplifying, you can eliminate unnecessary terms and focus on solving for the variable. A simplified equation not only looks cleaner but also reduces the margin for errors in further steps. This process is particularly useful when dealing with complex algebraic expressions.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify and solve equations. It states that \(a(b + c) = ab + ac\). This property becomes very handy when one needs to eliminate parentheses or expand expressions.
In applying it to the exercise, when multiplying \((3r - 1)\) with 4 on the right-hand side, it becomes \(12r - 4\). This expansion is crucial to solve for the variable without altering the equation's balance.

• Apply the distributive property to eliminate parentheses and distribute coefficients.
• Ensure to distribute every term inside the bracket.

Understanding and correctly applying the distributive property can greatly enhance your ability to manipulate and solve equations efficiently. It's a tool often used in algebra when dealing with expressions that need expanding.