Problem 24 - Prealgebra

Question

What number do you divide by 6 to get \(-7 ?\)

Step-by-Step Solution

Answer

The number is -42.

Step 1: Define the problem

We need to find a number that, when divided by 6, gives -7 as a result. We can represent the unknown number by the variable \( x \), leading to the equation: \( \frac{x}{6} = -7 \).

Step 2: Set up the equation

From the given problem statement, we translated it into the equation \( \frac{x}{6} = -7 \). Our goal now is to solve for \( x \).

Step 3: Multiply both sides by 6

To isolate \( x \), multiply both sides of the equation by 6: \( x = -7 \times 6 \).

Step 4: Calculate the product

Now, compute \(-7 \times 6\): \(-7 \times 6 = -42\).

Step 5: Conclude the solution

The unknown number \( x \) is equal to \(-42\), as dividing \(-42\) by 6 gives -7.

Key Concepts

Division Equations

Division equations involve dividing a variable by a number to get a specific result. Consider this like sharing something evenly. For instance, in the given problem, we're looking at the equation \( \frac{x}{6} = -7 \). Imagine that \( x \) is a number that, when split into 6 equal parts, each part equals \(-7\). Division equations often appear in problems where you need to find an unknown number that fits this kind of pattern.

Understanding division in equations is crucial because it frequently involves reversing the division to simplify or solve it. In other words, once you have a division equation, the next step is often to "undo" the division by multiplying. This is key to solving these types of equations quickly and effectively.

Solving for Variables

To solve for a variable means finding the value that makes an equation true. In our problem, the variable is \( x \). The goal is to determine what number \( x \) can be, so that when it's divided by 6, the result is \(-7\). First, write down the equation. It's \( \frac{x}{6} = -7 \).

Here's a simple way to approach solving for \( x \):

• Identify the operation performed on \( x \). Here, \( x \) is divided by 6.
• Use the inverse operation to isolate \( x \). The inverse of division is multiplication. So, multiply both sides by 6.

This gives us \( x = -7 \times 6 \).

By solving for \( x \), you determine what value makes the original equation accurate.

Negative Numbers

Dealing with negative numbers in equations is important, as they change outcomes. In the equation \( \frac{x}{6} = -7 \), the \(-7\) indicates we are working with a negative result. This means our solution should also reflect a negative when multiplied or calculated.

When multiplying \(-7\) and 6 in our equation, the product \(-42\) maintains the negative sign. This causes the division of \(-42\) by 6 to accurately reflect \(-7\). It's crucial to remember:

• A negative number multiplied by a positive number results in a negative.
• Two negative numbers multiplied give a positive result.

These rules will help you manage negative numbers effectively within equations.

Basic Equations

Basic equations are foundational math tools that balance both sides of an equation with unknown variables. In our example, the equation \( \frac{x}{6} = -7 \) is a basic equation because it includes a straightforward relationship where one operation is performed on \( x \). By understanding this, you'll easily comprehend and work through simple math problems.

The principle of balance is key. Whatever operation you perform on one side of the equation must be done to the other side to keep it equal. This is why multiplying both sides by 6 was essential. It preserved the balance while shifting the equation towards solving for \( x \).

Mastering these basic concepts in equations helps build a solid understanding for tackling more complex mathematical challenges later on.