The First And Second Steps To Solve The Equation $\frac{3x}{5} + 5 = 20$ Are Shown Below:Step 1: $\frac{3x}{5} + 5 - 5 = 20 - 5$Step 2: $\frac{3x}{5} \cdot \frac{5}{3} = 15 \cdot \frac{5}{3}$Which Property Was Applied In The

The Power of Algebra: Unraveling the Mystery of the Equation

Algebra, a branch of mathematics that deals with solving equations and manipulating variables, is a fundamental subject that has been a cornerstone of mathematics for centuries. It is a powerful tool that helps us solve problems in various fields, from science and engineering to economics and finance. In this article, we will delve into the world of algebra and explore the first and second steps to solve the equation $\frac{3x}{5} + 5 = 20$. We will also discuss the property that was applied in the second step.

The Equation: A Challenge to be Solved

The equation $\frac{3x}{5} + 5 = 20$ is a simple yet challenging equation that requires careful manipulation to solve. The equation involves a fraction, an integer, and a variable, making it a perfect example of an algebraic equation. Our goal is to isolate the variable $x$ and find its value.

Step 1: Simplifying the Equation

The first step to solve the equation is to simplify it by subtracting 5 from both sides. This is done to isolate the fraction $\frac{3x}{5}$ on one side of the equation.

$\frac{3x}{5} + 5 - 5 = 20 - 5$

By subtracting 5 from both sides, we get:

$\frac{3x}{5} = 15$

This simplification is a crucial step in solving the equation, as it allows us to focus on the fraction and the variable $x$.

Step 2: Applying the Multiplicative Inverse Property

The second step to solve the equation is to apply the multiplicative inverse property. This property states that if we have a fraction $\frac{a}{b}$, we can multiply both sides of the equation by the reciprocal of the fraction, which is $\frac{b}{a}$, to eliminate the fraction.

$\frac{3x}{5} \cdot \frac{5}{3} = 15 \cdot \frac{5}{3}$

By multiplying both sides of the equation by $\frac{5}{3}$, we get:

$x = 25$

This step is a critical part of solving the equation, as it allows us to eliminate the fraction and isolate the variable $x$.

The Multiplicative Inverse Property: A Key Concept in Algebra

The multiplicative inverse property is a fundamental concept in algebra that allows us to eliminate fractions and solve equations. This property is based on the idea that if we have a fraction $\frac{a}{b}$, we can multiply both sides of the equation by the reciprocal of the fraction, which is $\frac{b}{a}$, to eliminate the fraction.

Example: Applying the Multiplicative Inverse Property

Let's consider another example to illustrate the application of the multiplicative inverse property. Suppose we have the equation $\frac{2x}{4} = 6$. To solve this equation, we can apply the multiplicative inverse property by multiplying both sides of the equation by the reciprocal of the fraction, which is $\frac{4}{2}$.

$\frac{2x}{4} \cdot \frac{4}{2} = 6 \cdot \frac{4}{2}$

By multiplying both sides of the equation by $\frac{4}{2}$, we get:

$x = 12$

This example demonstrates how the multiplicative inverse property can be used to solve equations and eliminate fractions.

In conclusion, the first and second steps to solve the equation $\frac{3x}{5} + 5 = 20$ involve simplifying the equation and applying the multiplicative inverse property. The multiplicative inverse property is a fundamental concept in algebra that allows us to eliminate fractions and solve equations. By understanding and applying this property, we can solve a wide range of algebraic equations and unlock the secrets of mathematics.

Algebra is a powerful tool that has numerous applications in various fields. By mastering the concepts of algebra, we can solve problems and make informed decisions. The multiplicative inverse property is a key concept in algebra that allows us to eliminate fractions and solve equations. By understanding and applying this property, we can unlock the secrets of mathematics and achieve our goals.

• [1] "Algebra" by Michael Artin
• [2] "Introduction to Algebra" by Richard Rusczyk
• [3] "Algebra and Trigonometry" by James Stewart

• Multiplicative Inverse Property: A property that states that if we have a fraction $\frac{a}{b}$, we can multiply both sides of the equation by the reciprocal of the fraction, which is $\frac{b}{a}$, to eliminate the fraction.
• Reciprocal: The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$.
• Fraction: A fraction is a way of expressing a part of a whole as a ratio of two numbers.
• Variable: A variable is a symbol that represents a value that can change.
  Frequently Asked Questions: Algebra and the Multiplicative Inverse Property ====================================================================

Q: What is the multiplicative inverse property?

A: The multiplicative inverse property is a fundamental concept in algebra that states that if we have a fraction $\frac{a}{b}$, we can multiply both sides of the equation by the reciprocal of the fraction, which is $\frac{b}{a}$, to eliminate the fraction.

Q: How do I apply the multiplicative inverse property?

A: To apply the multiplicative inverse property, you need to identify the fraction in the equation and multiply both sides of the equation by the reciprocal of the fraction. For example, if you have the equation $\frac{2x}{4} = 6$, you can multiply both sides of the equation by $\frac{4}{2}$ to eliminate the fraction.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. For example, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

Q: Can I use the multiplicative inverse property with other types of equations?

A: Yes, you can use the multiplicative inverse property with other types of equations, such as linear equations and quadratic equations. However, you need to make sure that the equation is in the correct form and that you are applying the property correctly.

Q: What are some common mistakes to avoid when using the multiplicative inverse property?

A: Some common mistakes to avoid when using the multiplicative inverse property include:

• Not identifying the fraction in the equation
• Not multiplying both sides of the equation by the reciprocal of the fraction
• Not simplifying the equation after applying the property
• Not checking the solution to make sure it is correct

Q: How do I check my solution to make sure it is correct?

A: To check your solution, you need to plug the value back into the original equation and make sure that it is true. For example, if you have the equation $\frac{2x}{4} = 6$ and you solve for $x$ by multiplying both sides of the equation by $\frac{4}{2}$, you need to plug the value of $x$ back into the original equation to make sure that it is true.

Q: What are some real-world applications of the multiplicative inverse property?

A: The multiplicative inverse property has numerous real-world applications, including:

• Solving problems in physics and engineering
• Calculating interest rates and investments
• Determining the cost of goods and services
• Solving problems in computer science and programming

Q: Can I use the multiplicative inverse property with negative numbers?

A: Yes, you can use the multiplicative inverse property with negative numbers. However, you need to make sure that you are applying the property correctly and that you are not multiplying both sides of the equation by a negative number.

Q: What are some common misconceptions about the multiplicative inverse property?

A: Some common misconceptions about the multiplicative inverse property include:

• Thinking that the multiplicative inverse property only applies to fractions
• Thinking that the multiplicative inverse property only applies to linear equations
• Thinking that the multiplicative inverse property is only used in algebra
• Thinking that the multiplicative inverse property is a difficult concept to understand

In conclusion, the multiplicative inverse property is a fundamental concept in algebra that allows us to eliminate fractions and solve equations. By understanding and applying this property, we can solve a wide range of algebraic equations and unlock the secrets of mathematics.