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Chapter 1: Problem 10

Solve the given equations. $$\frac{x}{-4}=2$$

Short Answer

Expert verified
The solution is \( x = -8 \).

Step by step solution

01

Understand the equation

The equation given is \( \frac{x}{-4} = 2 \). It means \( x \) divided by \(-4\) equals 2. Our goal is to find the value of \( x \).
02

Isolate x

To isolate \( x \), multiply both sides of the equation by \(-4\) to cancel out the division by \(-4\). Thus, we have \( x = 2 \times (-4) \).
03

Calculate the result

Now calculate the right side: \( 2 \times (-4) = -8 \). Thus, \( x = -8 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Manipulation
Algebraic manipulation is the process we use to simplify or rearrange equations to make them easier to solve. In the example given, we started with the equation \( \frac{x}{-4} = 2 \). This is a basic algebraic equation that can be manipulated to isolate the variable \( x \). By performing operations on both sides of the equation, we manipulate it to deduce the value of \( x \).

Here's how you can think about it:
  • Identify the operation being performed on the variable.
  • Apply the opposite (inverse) operation to both sides to eliminate other terms around it.
  • Simplify the equation step by step to maintain balance between the two sides.
This strategy ensures we safely and precisely find the value of the unknown variable whenever we engage in algebraic manipulation.
Isolating Variables
Isolating variables often involves rearranging the components of an equation so that the variable in question stands alone on one side, making it clear what the variable equals. In the original exercise, the variable to isolate is \( x \). The equation \( \frac{x}{-4} = 2 \) has \( x \) currently divided by \(-4\).

To isolate \( x \), you need to reverse the current operation involving \( x \):
  • Since \( x \) is divided by \(-4\), you multiply by \(-4\) to eliminate the division.
  • Multiplying both sides of the equation by \(-4\) keeps the equation balanced.
  • The left side then simplifies to \( x \) when you perform the multiplication.
By isolating the variable, you find the exact value necessary to satisfy the equation, which is a crucial step in solving algebra problems.
Multiplication and Division of Integers
Understanding how to multiply and divide integers is essential when dealing with algebraic expressions and equations. In the exercise, the equation \( \frac{x}{-4} = 2 \) required us to multiply by \(-4\) to isolate \( x \). This involves understanding integer multiplication and division rules.

Here are the basic rules for handling integers:
  • When multiplying or dividing two numbers of like signs (both positive or both negative), the result is positive.
  • When multiplying or dividing two numbers of unlike signs (one positive, one negative), the result is negative.
In our case, \( 2 \times (-4) \) gives \(-8\) because we are multiplying a positive number and a negative number, resulting in a negative product. Knowing these rules helps efficiently manage and solve algebraic equations involving integers.

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Most popular questions from this chapter

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