Problem 112 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{3 z}{4}=\frac{-7}{8} $$

Step-by-Step Solution

Answer

The solution is \( z = \frac{-7}{6} \).

Step 1: Understand the Equation

We are given the equation \( \frac{3z}{4} = \frac{-7}{8} \). This is a simple equation involving a variable \( z \) and two fractions on either side of the equation.

Step 2: Eliminate the Fraction Coefficients

To eliminate the fractions, let's multiply both sides by the least common denominator of 4 and 8. In this case, the least common denominator is 8. Multiply both sides by 8: \[ 8 \times \frac{3z}{4} = 8 \times \frac{-7}{8} \]This gives us \( 6z = -7 \).

Step 3: Solve for z

Now that we have the equation \( 6z = -7 \), divide both sides by 6 to solve for \( z \): \[ z = \frac{-7}{6} \]. Hence, the solution to the equation is \( z = \frac{-7}{6} \).

Key Concepts

Fractions in Equations

When tackling fractions in equations, it's essential to understand how they operate in a mathematical equation. Fractions are numbers representing parts of a whole. Each fraction has a numerator (the top number) and a denominator (the bottom number). In equations, fractions often appear when dividing a variable or a number. For example, in the equation \( \frac{3z}{4} = \frac{-7}{8} \), both sides have a fraction.

Handling fractions in equations requires careful manipulation to keep both sides balanced. This often involves finding a common denominator or clearing the fractions entirely to simplify the equation. One effective strategy is to eliminate the fractions by multiplying through by the least common denominator (LCD).

• Make sure you handle both sides of the equation equally.
• Avoid making errors in simple arithmetic while dealing with fractions.

Understanding how fractions work in equations is a crucial step to solving them accurately.

Variables and Coefficients

In equations involving fractions, variables and coefficients play a crucial role. A variable represents an unknown value, often noted as \( z \), \( x \), or \( y \). Coefficients are the numbers multiplying the variable. In the equation \( \frac{3z}{4} = \frac{-7}{8} \), 3 is the coefficient of the variable \( z \).

Variables allow equations to represent more than one possibility and can be solved to find specific solutions. Coefficients indicate how many times to multiply the variable and they affect the balance of the equation. By isolating the variable, you can solve for its value.

• Identify the variable and its coefficient in the equation.
• Focus on isolating the variable to one side for solving purposes.

Whenever you encounter an equation, understanding the role of each variable and its coefficient is key to finding the solution.

Least Common Denominator

When solving equations with fractions, finding the least common denominator (LCD) simplifies the process. The LCD is the smallest number that both denominators of the fractions can divide into without a remainder. For the equation \( \frac{3z}{4} = \frac{-7}{8} \), the denominators are 4 and 8. The LCD here is 8 because it's the smallest number both can divide into evenly.

Once you've identified the LCD, multiply both sides of the equation by this value. This helps eliminate the fractions and transforms the equation into one with whole numbers or simpler fractions. For example:

• Multiply \( \frac{3z}{4} \) by 8 to get \( 6z \).
• Multiply \( \frac{-7}{8} \) by 8 to get \(-7\).

This method simplifies solving the equation and makes calculations more straightforward, helping you to easily find the value of the variable.