Problem 61 - Prealgebra

Question

PREREQUISITE SKILL Solve each equation. $$\frac{d}{-3}=-6$$

Step-by-Step Solution

Answer

The solution is \( d = 18 \).

Step 1: Understand the Equation

The equation is given as \( \frac{d}{-3} = -6 \). This means the variable \( d \) has been divided by \(-3\), and the result of this division is equal to \(-6\). To find \( d \), we need to eliminate the division by \(-3\).

Step 2: Isolate the Variable

To solve for \( d \), multiply both sides of the equation by \(-3\) to cancel out the division. The equation becomes:\[ d = -6 \times (-3) \]

Step 3: Simplify the Right Side

Calculate \(-6 \times (-3)\). When two negative numbers are multiplied, the result is positive. Thus, \(-6 \times (-3) = 18\).

Step 4: Write the Final Answer

Thus, the value of \( d \) is \( 18 \). The solution to the equation \( \frac{d}{-3} = -6 \) is \( d = 18 \).

Key Concepts

Understanding Mathematical Operations

Mathematical operations are the steps we use to solve equations. They include addition, subtraction, multiplication, and division. In our exercise, the key operation is division. We start with the equation \( \frac{d}{-3} = -6 \). This tells us that the variable \( d \) is being divided by \(-3\).
To solve the equation, we perform the opposite operation to eliminate the division. In this case, we multiply both sides of the equation by \(-3\). By doing so, we aim to "undo" the division, which helps us understand the importance of inverse operations in solving equations. This highlights how we can manipulate equations to make them easier to solve.

• Division can be reversed by multiplication.
• Always use the opposite operation to isolate the variable.

Understanding these operations allows us to handle equations dynamically. It equips us with tools to simplify and solve them efficiently.

Handling Negative Numbers

Working with negative numbers can sometimes be tricky, but it's crucial to solving equations accurately. In the equation \( \frac{d}{-3} = -6 \), both the divisor and the result are negative numbers.
When multiplying or dividing with negative numbers, important rules apply:

• Multiplying or dividing two negative numbers results in a positive number.
• Multiplying or dividing a positive number by a negative number gives a negative result.

Applying these rules to our exercise: when we multiplied \(-6\) by \(-3\), we got \(18\). This positive result illustrates the rule of handling negatives efficiently. Remember, keeping these rules in mind during operations is critical to maintaining accuracy.
Negative numbers are everywhere, and understanding these simple rules can greatly enhance your problem-solving abilities.

Mastering Variable Isolation

Variable isolation is a fundamental concept in algebra. It involves rearranging an equation to express one variable in terms of others. The main goal is to get the variable alone on one side of the equation.
In our exercise, the equation begins as \( \frac{d}{-3} = -6 \). To isolate \( d \), we performed the inverse operation of division, which is multiplication. By multiplying both sides by \(-3\), we effectively canceled out the divisor on the left side. This left us with a simple expression: \( d = 18 \).

• Identify the operation currently affecting the variable.
• Apply the inverse operation to both sides of the equation.

Once isolated, the variable's value becomes clear and the solution is obvious. Mastering variable isolation means you're one step closer to understanding more complex equations, enhancing your mathematical reasoning and confidence.