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Chapter 2: Problem 39

Solve the equations by using the multiplication property. $$\frac{x}{5}=0$$

Short Answer

Expert verified
The solution is \( x = 0 \).

Step by step solution

01

Understand the Equation

The given equation is \( \frac{x}{5} = 0 \). This means that when \( x \) is divided by 5, the result is 0.
02

Apply the Multiplication Property

According to the multiplication property of equality, we can multiply both sides of the equation by the same non-zero number. To eliminate the fraction, multiply both sides by 5: \( 5 \times \frac{x}{5} = 5 \times 0 \).
03

Simplify the Equation

The multiplication on the left side simplifies as follows: \( 5 \times \frac{x}{5} = x \). On the right side, \( 5 \times 0 = 0 \). Hence, the equation simplifies to \( x = 0 \).

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Key Concepts

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

Solving Equations
Solving equations involves finding the value of the unknown variable that satisfies the equation. In this exercise, we're working with an equation in the form of a fraction where the variable, \( x \), is part of the numerator and a number divides it. The goal is to isolate the variable by performing operations on both sides of the equation that maintain its balance. This means whatever you do to one side of the equation, you must do to the other. By applying the multiplication property of equality, we ensure that our manipulations do not affect the equality of the equation. This principle is vital in algebra as it allows us to solve for variables step by step.
Fraction Elimination
Fraction elimination is a key step when solving equations that involve fractions. The aim is to get rid of the fraction in order to simplify the equation. In our exercise, the equation is \( \frac{x}{5} = 0 \). To eliminate the fraction, you multiply both sides by the denominator鈥攊n this case, 5. This is because multiplying by the denominator cancels the fraction:
  • On the left side, \( 5 \times \frac{x}{5} \) simplifies to \( x \), because the 5s cancel each other out.
  • On the right side, \( 5 \times 0 \) equals 0, leaving us with the equation \( x = 0 \).
This step is crucial as it transforms a potentially complex equation into a simpler form, making it easier to find the solution.
Basic Algebra Concepts
Understanding basic algebra concepts is foundational for solving equations effectively. At its core, algebra deals with variables, numbers, and the rules for manipulating them. When you encounter a problem like \( \frac{x}{5} = 0 \), it's important to recognize the need to operate on both sides of the equation equally. Here are some key points related to basic algebra concepts that play a part in this exercise:
  • **Equality Principle**: Whatever operation you perform on one side of an equation, you must perform on the other side to maintain balance.
  • **Variable Isolation**: The ultimate goal is to get the variable (in this case, \( x \)) alone on one side of the equation, providing a straightforward solution.
  • **Operations on Zero**: Remember that any number multiplied by zero remains zero. This is helpful in verifying our final answer.
These concepts are the building blocks of effectively tackling more complex algebraic problems, and mastering them will aid in solving a wide range of mathematical equations.

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Most popular questions from this chapter

Solve the equations by using the division property. $$5 x=-45$$

Consider a variable \(x\) where \(x\) represents the whole numbers from 0 to 20. Stated mathematically, the possible values of \(x\) are $$x=0,1,2, \ldots, 19,20$$ For each of the phrases, write an inequality and then determine which values satisfy that inequality. \(x\) is at least 14

Solve the equations by using the division property. $$2 x=12$$

Solve the equations by using the multiplication property. $$\frac{y}{5}=6$$

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