Problem 20 - Fundamentals of Mathematics

Question

Solve each equation. $$ \frac{7 y}{8}+\frac{1}{4}=\frac{-13}{4} $$

Step-by-Step Solution

Answer

The solution is \( y = -4 \).

Step 1: Eliminate Fractions

To eliminate fractions, multiply every term in the equation by the least common denominator, which in this case is 8. So, multiply both sides of the equation by 8:\[ 8 \left( \frac{7y}{8} \right) + 8 \left( \frac{1}{4} \right) = 8 \left( \frac{-13}{4} \right) \]Simplifying each term gives:\[ 7y + 2 = -26 \]

Step 2: Isolate the Variable

Subtract 2 from both sides of the equation to isolate the term with the variable:\[ 7y + 2 - 2 = -26 - 2 \]This simplifies to:\[ 7y = -28 \]

Step 3: Solve for the Variable

Divide both sides of the equation by 7 to solve for \( y \):\[ \frac{7y}{7} = \frac{-28}{7} \]Simplifying both sides gives:\[ y = -4 \]

Key Concepts

Eliminating Fractions

When faced with an equation that includes fractions, the first step is often to eliminate them. This simplifies the equation and makes it easier to work with. In mathematical terms, each term is multiplied by the least common denominator (LCD) to clear the fractions.

Imagine you are dealing with different pieces of a puzzle that don't quite fit together because they aren't in the same format. This is exactly what fractions do in equations, and by finding the LCD, we establish a common ground. In our original equation, \( \frac{7y}{8} + \frac{1}{4} = \frac{-13}{4} \), the least common denominator is 8.

Here are the major steps to eliminate fractions:

• Identify the LCD among all fractions in the equation.
• Multiply every term in the equation by this LCD.

For our equation, multiplying every term by 8 results in:

\[ 8 \left( \frac{7y}{8} \right) + 8 \left( \frac{1}{4} \right) = 8 \left( \frac{-13}{4} \right) \]

This step effectively clears the fractions, simplifying the equation to \( 7y + 2 = -26 \). Notice how the puzzle pieces now fit more neatly together.

Isolating the Variable

Once you have simplified your equation by eliminating fractions, the next move is to isolate the variable. This is like peeling away layers to reveal the core. You want to end up with the variable on one side and the numbers on the other.

To isolate the variable in an equation like \( 7y + 2 = -26 \), you need to perform inverse operations. Here's how it works:

• Look for terms added to or subtracted from the variable.
• Use subtraction or addition to eliminate these terms from one side of the equation.

For the equation \( 7y + 2 = -26 \), you subtract 2 from both sides to keep the equation balanced:

\[ 7y + 2 - 2 = -26 - 2 \]

This results in \( 7y = -28 \), effectively isolating the term with the variable \( y \) on one side.

Isolating the variable helps to narrow down the focus of solving the equation, like zooming in on the most critical piece of information you need.

Least Common Denominator

The least common denominator (LCD) is a crucial helper in dealing with equations that contain fractions. Think of it as a common language that allows various parts of your equation to communicate effortlessly. Without it, fractions may cause confusion like speaking in different tongues.

The LCD is the smallest number that all the denominators of the fractions can divide into evenly. When you find the LCD, multiply every term in your equation by it to eliminate the fractions.

In our initial equation, \( \frac{7y}{8} + \frac{1}{4} = \frac{-13}{4} \), we notice that 8 and 4 are the denominators. The number 8 is the least number that both can divide into without a remainder. Hence, it is the LCD in this case.

Key points to remember about the LCD:

• It's all about finding the smallest number that denominators share as a multiple.
• This step is essential for simplifying equations and making them easier to solve.

Once you've found and used the LCD, the fractions vanish, allowing you to work with a simpler, more straightforward equation.