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Chapter 4: Problem 30

Solve the equations. Write the answers as fractions or whole numbers. $$-\frac{v}{10}=-4$$

Short Answer

Expert verified
The solution is \( v = 40 \).

Step by step solution

01

Understand the Equation

The equation to solve is \(-\frac{v}{10} = -4\). The goal is to isolate the variable \(v\).
02

Eliminate the Negative Sign

Since both sides of the equation are negative, multiply both sides by \(-1\) to remove the negative sign: \[ -1 \times -\frac{v}{10} = -1 \times -4 \] This simplifies to: \[ \frac{v}{10} = 4 \]
03

Solve for v

To solve for \(v\), multiply both sides of the equation by 10 to isolate \(v\): \[ 10 \times \frac{v}{10} = 10 \times 4 \] This simplifies to: \[ v = 40 \]

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Key Concepts

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

Isolate the Variable
To solve equations, one of the main goals is to isolate the variable. This means getting the variable by itself on one side of the equation. In the given equation, \(-\frac{v}{10} = -4\), the variable is \(v\). By isolating \(v\), we can determine its value. Here we describe how to achieve the isolation step by step:
  • First, identify the variable you need to isolate (in this case, \(v\)).
  • Then, perform operations to remove any numbers, fractions, or coefficients attached to the variable. This often includes the inverse operations like adding, subtracting, multiplying, or dividing.
Remember, whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced. In our example, we will be manipulating both sides equally to isolate \(v\).
Eliminate Negative Sign
When solving equations, encountering negative signs can complicate things. Removing negative signs simplifies the equation. Given the equation \(-\frac{v}{10} = -4\), observe the negative sign on both sides. Here鈥檚 how to eliminate them:
  • Identify both sides of the equation that have negative signs.
  • Multiply each side by \(-1\). This converts negative numbers to positive numbers and vice versa. In our example, multiplying yields: \[ -1 \times -\frac{v}{10} = -1 \times -4 \] which simplifies to \[ \frac{v}{10} = 4 \]
  • After eliminating the negative signs, proceed with isolating the variable further.
Removing negatives simplifies further multiplication and division steps needed to isolate variables.
Multiplication in Equations
After addressing negative signs and fractions, multiplication is often used to isolate variables completely. In \( \frac{v}{10} = 4 \), we need to eliminate the fraction by multiplying both sides of the equation by 10. Here鈥檚 a simple explanation:
  • Identify the denominator that is attached to the variable (here it is 10).
  • Multiply both sides by this denominator to eliminate the fraction. For this equation, multiply 10 by both sides: \[ 10 \times \frac{v}{10} = 10 \times 4 \]
This step simplifies to: \[ v = 40 \]
Multiplying both sides by the same number helps keep the equation balanced, simplifying the terms, and isolating the variable.

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