Problem 8 - Elementary Algebra

Question

$$ \text { For Problems 1-32, solve each equation. (Objective 1) } $$ $$ \frac{2}{3 t}+\frac{3}{4 t}=1-\frac{5}{2 t} $$

Step-by-Step Solution

Answer

The solution to the equation is \(t = \frac{47}{12}\).

Step 1: Identify the Least Common Denominator (LCD)

To begin solving, we first identify the least common denominator (LCD) of all the fractions in terms of \(t\). The denominators are \(3t\), \(4t\), and \(2t\). The LCD for these is \(12t\).

Step 2: Clear the Fractions

Multiply every term in the equation by the LCD, \(12t\), to eliminate the fractions. This gives:\[ 12t \left( \frac{2}{3t} \right) + 12t \left( \frac{3}{4t} \right) = 12t(1) - 12t \left( \frac{5}{2t} \right) \] Simplifying, we get: \[ 8 + 9 = 12t - 30 \]

Step 3: Simplify the Equation

Combine the constants on the left side: \[ 17 = 12t - 30 \] Next, add 30 to both sides to isolate terms with \(t\) on one side:\[ 47 = 12t \]

Step 4: Solve for t

Divide both sides by 12 to solve for \(t\): \[ t = \frac{47}{12} \] This fraction is in its simplest form.

Key Concepts

Understanding the Least Common Denominator

When solving equations that involve fractions, it's often necessary to find a common denominator for all the fractional terms.
In this exercise, we look at the denominators of the fractions: \(3t\), \(4t\), and \(2t\). The goal is to find the least common denominator (LCD).
The LCD is the smallest expression that all the denominators can divide into without leaving a remainder.

• Consider the coefficients: 3, 4, and 2. The least common multiple of these numbers is 12.
• Incorporate the variable \(t\): Since each term has \(t\) as part of its denominator, the LCD will be \(12t\).

Finding the LCD helps in simplifying the equation, as it allows you to eliminate fractions by multiplying each term of the equation by this common denominator. This step is crucial in making the equation easier to solve.

Solving Fractions in Equations

Once the least common denominator is determined, we can proceed to eliminate the fractions from the equation to simplify it.
Multiplying every term in the equation by the LCD is a method called 'clearing the fractions'.
Let's explore this with our exercise:

• Original equation: \(\frac{2}{3t} + \frac{3}{4t} = 1 - \frac{5}{2t}\)
• After multiplying by \(12t\), it becomes \(12t\left(\frac{2}{3t}\right) + 12t\left(\frac{3}{4t}\right) = 12t - 12t\left(\frac{5}{2t}\right)\).

This simplifies to integers and constants: \(8 + 9 = 12t - 30\).
The process removes fractions, making it straightforward to handle, and transforms it into a linear equation which is easier to solve.

Understanding Linear Equations

A linear equation like \(17 = 12t - 30\) is straightforward. It involves variables to the first power and forms a straight line when graphed.
Linear equations can be solved through basic algebraic operations like addition, subtraction, multiplication, and division.
For our exercise, follow these steps:

• Combine like terms: Simplify \(8 + 9\) to get \(17\).
• Isolate the variable \(t\): Add 30 to both sides to get \(47 = 12t\).
• Solve for \(t\): Divide both sides by 12, resulting in \(t = \frac{47}{12}\).

This result, \(\frac{47}{12}\), is the simplest form of the solution for \(t\).
Understanding these steps, especially in dealing with linear equations, is fundamental to solving similar algebraic problems efficiently.