Marina Is Solving The Equation Below By Using The Multiplicative Inverse, Or The Reciprocal.$\[\frac{1}{5}(x-2) = 20\\]By Which Number Should Marina Multiply Both Sides Of The Equation As The First Step In Solving It?A.

Feb 28, 2025 by ADMIN 220 views

**Solving Equations with Multiplicative Inverse: A Step-by-Step Guide**

Introduction

In mathematics, solving equations is a crucial skill that involves manipulating algebraic expressions to isolate the variable. One of the techniques used to solve equations is the multiplicative inverse, also known as the reciprocal. In this article, we will explore how to use the multiplicative inverse to solve equations, using the example of Marina, who is trying to solve the equation $\frac{1}{5}(x-2) = 20$ .

Understanding the Multiplicative Inverse

The multiplicative inverse of a number is its reciprocal, which is obtained by flipping the number. For example, the multiplicative inverse of 5 is $\frac{1}{5}$ , and the multiplicative inverse of 2 is $\frac{1}{2}$ . When we multiply a number by its multiplicative inverse, we get 1.

Applying the Multiplicative Inverse to the Equation

Now, let's apply the multiplicative inverse to the equation $\frac{1}{5}(x-2) = 20$ . To do this, we need to multiply both sides of the equation by the multiplicative inverse of $\frac{1}{5}$ , which is 5.

Step 1: Multiply Both Sides by 5

To solve the equation, Marina needs to multiply both sides by 5, which is the multiplicative inverse of $\frac{1}{5}$ . This will eliminate the fraction on the left-hand side of the equation.

\frac{1}{5}(x-2) = 20
\implies 5 \times \frac{1}{5}(x-2) = 5 \times 20
\implies x-2 = 100

Step 2: Add 2 to Both Sides

Now that we have eliminated the fraction, we can add 2 to both sides of the equation to isolate the variable x.

x-2 = 100
\implies x-2 + 2 = 100 + 2
\implies x = 102

Conclusion

In this article, we have seen how to use the multiplicative inverse to solve equations. By multiplying both sides of the equation by the multiplicative inverse of $\frac{1}{5}$ , Marina was able to eliminate the fraction and solve for the variable x. This technique can be applied to a wide range of equations, and it is an essential tool for any mathematician.

Tips and Tricks

When using the multiplicative inverse, make sure to multiply both sides of the equation by the correct value.
Be careful when adding or subtracting values from both sides of the equation.
Use the multiplicative inverse to eliminate fractions and simplify equations.

Real-World Applications

The concept of multiplicative inverse has many real-world applications, including:

Finance: When calculating interest rates or investment returns, the multiplicative inverse is used to determine the reciprocal of a value.
Science: In physics and engineering, the multiplicative inverse is used to calculate the reciprocal of a value, such as the reciprocal of a distance or a time.
Computer Science: In programming, the multiplicative inverse is used to calculate the reciprocal of a value, such as the reciprocal of a number or a fraction.

Common Mistakes

Failing to multiply both sides of the equation by the correct value.
Adding or subtracting values from only one side of the equation.
Not simplifying the equation after using the multiplicative inverse.

Conclusion

Q: What is the multiplicative inverse?

A: The multiplicative inverse of a number is its reciprocal, which is obtained by flipping the number. For example, the multiplicative inverse of 5 is $\frac{1}{5}$ , and the multiplicative inverse of 2 is $\frac{1}{2}$ .

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

A: To find the multiplicative inverse of a number, you simply need to flip the number. For example, the multiplicative inverse of 5 is $\frac{1}{5}$ , and the multiplicative inverse of 2 is $\frac{1}{2}$ .

Q: What is the purpose of the multiplicative inverse in solving equations?

A: The multiplicative inverse is used to eliminate fractions and simplify equations. By multiplying both sides of the equation by the multiplicative inverse of a fraction, you can eliminate the fraction and solve for the variable.

Q: How do I use the multiplicative inverse to solve an equation?

A: To use the multiplicative inverse to solve an equation, you need to multiply both sides of the equation by the multiplicative inverse of the fraction. This will eliminate the fraction and allow you to solve for the variable.

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

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

Failing to multiply both sides of the equation by the correct value.
Adding or subtracting values from only one side of the equation.
Not simplifying the equation after using the multiplicative inverse.

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

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

Finance: When calculating interest rates or investment returns, the multiplicative inverse is used to determine the reciprocal of a value.
Science: In physics and engineering, the multiplicative inverse is used to calculate the reciprocal of a value, such as the reciprocal of a distance or a time.
Computer Science: In programming, the multiplicative inverse is used to calculate the reciprocal of a value, such as the reciprocal of a number or a fraction.

Q: Can I use the multiplicative inverse to solve equations with variables on both sides?

A: Yes, you can use the multiplicative inverse to solve equations with variables on both sides. However, you need to be careful when multiplying both sides of the equation by the multiplicative inverse, as this can lead to errors.

Q: How do I know when to use the multiplicative inverse to solve an equation?

A: You should use the multiplicative inverse to solve an equation when the equation contains a fraction and you want to eliminate the fraction. This will allow you to simplify the equation and solve for the variable.

Q: Can I use the multiplicative inverse to solve equations with decimals?

A: Yes, you can use the multiplicative inverse to solve equations with decimals. However, you need to be careful when multiplying both sides of the equation by the multiplicative inverse, as this can lead to errors.

Q: How do I simplify an equation after using the multiplicative inverse?

A: To simplify an equation after using the multiplicative inverse, you need to multiply both sides of the equation by the multiplicative inverse of the fraction. This will eliminate the fraction and allow you to solve for the variable.

Q: Can I use the multiplicative inverse to solve equations with negative numbers?

A: Yes, you can use the multiplicative inverse to solve equations with negative numbers. However, you need to be careful when multiplying both sides of the equation by the multiplicative inverse, as this can lead to errors.

Conclusion

In conclusion, the multiplicative inverse is a powerful tool for solving equations. By understanding how to use the multiplicative inverse, mathematicians can simplify equations and solve for variables. This technique has many real-world applications and is an essential tool for any mathematician.