Problem 7 - Elementary Algebra

Question

$$ \text { For Problems 1-32, solve each equation. (Objective 1) } $$ $$ \frac{3}{2 t}-\frac{5}{t}=\frac{7}{5 t}+1 $$

Step-by-Step Solution

Answer

The solution is \( t = -\frac{49}{10} \).

Step 1: Identify the lowest common denominator (LCD)

The first step in solving the equation \( \frac{3}{2t} - \frac{5}{t} = \frac{7}{5t} + 1 \) is to identify the lowest common denominator for all the terms. The denominators in the equation are \( 2t \), \( t \), and \( 5t \). The lowest common denominator that includes all these terms is \( 10t \).

Step 2: Multiply each term by the LCD

Next, we multiply every term in the equation by the LCD \( 10t \) to eliminate the fractions. This results in: \[ 10t \times \left( \frac{3}{2t} \right) - 10t \times \left( \frac{5}{t} \right) = 10t \times \left( \frac{7}{5t} \right) + 10t \times 1 \] Simplifying, this gives: \[ 15 - 50 = 14 + 10t \]

Step 3: Simplify the equation

We simplify the equation resulting from Step 2: \[ 15 - 50 = 14 + 10t \] Which simplifies to: \[ -35 = 14 + 10t \]

Step 4: Solve for \( t \)

Re-arrange the equation to isolate \( t \): \[ -35 - 14 = 10t \] Which simplifies to: \[ -49 = 10t \] Now, divide both sides by \( 10 \) to solve for \( t \): \[ t = -\frac{49}{10} \]

Key Concepts

Fractions in Algebra

When dealing with fractions in algebra, it's important to remember that they are not much different from the fractions you encounter in arithmetic. Each fraction consists of a numerator (the top part) and a denominator (the bottom part). In algebra, variables can appear in the numerators, denominators, or both.

To work effectively with algebraic fractions, here are some key points to consider:

• Fractions represent division, so always think of the fraction bar as a division operation.
• Keep track of the variable within the fraction, as it can change the entire expression.
• Simplify fractions by canceling common factors in the numerator and denominator when possible.

In our given problem, you can see that each term within the equation contains a fraction. The challenge is to manipulate these fractions to solve for the variable 't.' This is achieved by finding a way to eliminate the denominators and simplify the equation.

Lowest Common Denominator

Before you can conveniently work with fractional equations, it's crucial to find the lowest common denominator (LCD). This process is akin to finding a common baseline for all fractions involved, enabling straightforward manipulation.

Here's how to efficiently find the LCD:

• Identify all denominators in the equations. In our problem, the denominators are \( 2t, t, \) and \( 5t \).
• Determine the least number that each denominator can divide into without leftovers. For \( 2t, t, \) and \( 5t \), this is \( 10t \).

Using the LCD, you can multiply each fraction, effectively "clearing" the denominators. This step turns a complex-looking problem with fractions into one with whole numbers, which are easier to manage. By multiplying each term in the equation \( \frac{3}{2t} - \frac{5}{t} = \frac{7}{5t} + 1 \) by \( 10t \), the fractions vanish, simplifying our work.

Isolate the Variable

Once the equation is free of fractions, isolating the variable becomes the main goal. This means getting the variable by itself on one side of the equation, so you can find its value.

Here's a step-by-step approach to isolate a variable like 't':

• Simplify the equation where possible by combining like terms. In the example, simplify \( 15 - 50 = 14 + 10t \) to \( -35 = 14 + 10t \).
• Move all terms with the variable to one side of the equation and constant terms to the other. Here, you'd subtract 14 from both sides, yielding \( -35 - 14 = 10t \).
• Simplify further to \( -49 = 10t \).
• Finally, divide by the coefficient of the variable to solve for it. For 't', divide both sides by 10, concluding with \( t = -\frac{49}{10} \).

This is the essence of solving any equation: simplifying and rearranging until the unknown variable is successfully isolated.