See The Solution Below:$\[ \begin{array}{l} \frac{7}{3} X = -2 \\ \frac{7}{3} \times \left(\frac{3}{7}\right) = -2 \times \left(\frac{3}{7}\right) \\ x = -\frac{6}{8} \end{array} \\]What Property Was Used To Solve The Equation?A. Property Of

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Introduction

In mathematics, solving equations is a crucial skill that involves manipulating algebraic expressions to isolate the variable. One of the fundamental properties used to solve equations is the multiplicative inverse property. In this article, we will explore how to use the multiplicative inverse property to solve equations and provide examples to illustrate the concept.

What is the Multiplicative Inverse Property?

The multiplicative inverse property states that for any non-zero number a, there exists a number b such that a * b = 1. This property is also known as the reciprocal property. In other words, the multiplicative inverse of a number a is a number b that, when multiplied by a, results in 1.

How to Use the Multiplicative Inverse Property to Solve Equations

To use the multiplicative inverse property to solve equations, we need to follow these steps:

  1. Isolate the variable: The first step is to isolate the variable on one side of the equation. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
  2. Identify the multiplicative inverse: Once the variable is isolated, we need to identify the multiplicative inverse of the coefficient of the variable. The multiplicative inverse of a number a is a number b such that a * b = 1.
  3. Multiply both sides by the multiplicative inverse: Once we have identified the multiplicative inverse, we need to multiply both sides of the equation by the multiplicative inverse. This will result in the variable being isolated on one side of the equation.

Example 1: Solving an Equation with the Multiplicative Inverse Property

Let's consider the equation 7/3 * x = -2. To solve this equation, we need to isolate the variable x. We can do this by multiplying both sides of the equation by the multiplicative inverse of 7/3, which is 3/7.

\frac{7}{3} * x = -2
\frac{7}{3} * \left(\frac{3}{7}\right) = -2 * \left(\frac{3}{7}\right)
x = -\frac{6}{8}

In this example, we used the multiplicative inverse property to solve the equation. We isolated the variable x by multiplying both sides of the equation by the multiplicative inverse of 7/3, which is 3/7.

Example 2: Solving an Equation with the Multiplicative Inverse Property

Let's consider the equation 4/5 * x = 3. To solve this equation, we need to isolate the variable x. We can do this by multiplying both sides of the equation by the multiplicative inverse of 4/5, which is 5/4.

\frac{4}{5} * x = 3
\frac{4}{5} * \left(\frac{5}{4}\right) = 3 * \left(\frac{5}{4}\right)
x = \frac{15}{4}

In this example, we used the multiplicative inverse property to solve the equation. We isolated the variable x by multiplying both sides of the equation by the multiplicative inverse of 4/5, which is 5/4.

Conclusion

In conclusion, the multiplicative inverse property is a fundamental concept in mathematics that is used to solve equations. By isolating the variable and identifying the multiplicative inverse of the coefficient, we can use the multiplicative inverse property to solve equations. The examples provided in this article illustrate how to use the multiplicative inverse property to solve equations.

Frequently Asked Questions

Q: What is the multiplicative inverse property?

A: The multiplicative inverse property states that for any non-zero number a, there exists a number b such that a * b = 1.

Q: How to use the multiplicative inverse property to solve equations?

A: To use the multiplicative inverse property to solve equations, we need to isolate the variable on one side of the equation, identify the multiplicative inverse of the coefficient, and multiply both sides of the equation by the multiplicative inverse.

Q: What is the multiplicative inverse of a number?

A: The multiplicative inverse of a number a is a number b such that a * b = 1.

Q: Can the multiplicative inverse property be used to solve equations with fractions?

Q: What is the multiplicative inverse property?

A: The multiplicative inverse property states that for any non-zero number a, there exists a number b such that a * b = 1. This property is also known as the reciprocal property.

Q: How to use the multiplicative inverse property to solve equations?

A: To use the multiplicative inverse property to solve equations, you need to follow these steps:

  1. Isolate the variable: Isolate the variable on one side of the equation.
  2. Identify the multiplicative inverse: Identify the multiplicative inverse of the coefficient of the variable.
  3. Multiply both sides by the multiplicative inverse: Multiply both sides of the equation by the multiplicative inverse.

Q: What is the multiplicative inverse of a number?

A: The multiplicative inverse of a number a is a number b such that a * b = 1. For example, the multiplicative inverse of 2 is 1/2, because 2 * 1/2 = 1.

Q: Can the multiplicative inverse property be used to solve equations with fractions?

A: Yes, the multiplicative inverse property can be used to solve equations with fractions. For example, if you have the equation 1/2 * x = 3, you can use the multiplicative inverse property to solve for x.

Q: How do I find the multiplicative inverse of a fraction?

A: To find the multiplicative inverse of a fraction, you need to flip the numerator and denominator. For example, the multiplicative inverse of 1/2 is 2/1, because 1/2 * 2/1 = 1.

Q: Can the multiplicative inverse property be used to solve equations with decimals?

A: Yes, the multiplicative inverse property can be used to solve equations with decimals. For example, if you have the equation 0.5 * x = 3, you can use the multiplicative inverse property to solve for x.

Q: How do I find the multiplicative inverse of a decimal?

A: To find the multiplicative inverse of a decimal, you need to convert the decimal to a fraction and then flip the numerator and denominator. For example, the multiplicative inverse of 0.5 is 2/1, because 0.5 = 1/2 and 1/2 * 2/1 = 1.

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 isolating the variable: Make sure to isolate the variable on one side of the equation before using the multiplicative inverse property.
  • Not identifying the multiplicative inverse: Make sure to identify the multiplicative inverse of the coefficient of the variable.
  • Not multiplying both sides by the multiplicative inverse: Make sure to multiply both sides of the equation by the multiplicative inverse.

Q: How can I practice using the multiplicative inverse property?

A: You can practice using the multiplicative inverse property by working through examples and exercises. You can also try solving equations with fractions and decimals to get a feel for how the property works.

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

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

  • Finance: The multiplicative inverse property is used in finance to calculate interest rates and investment returns.
  • Science: The multiplicative inverse property is used in science to calculate rates of change and proportions.
  • Engineering: The multiplicative inverse property is used in engineering to calculate stress and strain on materials.

Conclusion

In conclusion, the multiplicative inverse property is a fundamental concept in mathematics that is used to solve equations. By understanding how to use the multiplicative inverse property, you can solve equations with fractions and decimals and apply the property to real-world problems.