Multiply both sides by the LCM

Oct 13, 2025 by talynt.org 20 views

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Introduction

In algebra, solving equations is a crucial skill that involves manipulating mathematical expressions to isolate the variable of interest. One of the techniques used to solve equations is cross-products, which is a method of solving linear equations by multiplying both sides of the equation by a common factor. In this article, we will explore how to solve for $x$ using cross-products, focusing on the equation $\frac{18}{x} = \frac{x}{32}$ .

What are Cross-Products?

Cross-products is a technique used to solve linear equations by multiplying both sides of the equation by a common factor. This technique is based on the concept of multiplying both sides of an equation by a common factor to eliminate the fractions. The common factor is usually the least common multiple (LCM) of the denominators of the fractions.

How to Solve for $x$ using Cross-Products

To solve for $x$ using cross-products, we need to follow these steps:

Write the equation: Write the equation that we want to solve, in this case, $\frac{18}{x} = \frac{x}{32}$ .
Multiply both sides by the LCM: Multiply both sides of the equation by the LCM of the denominators, which is $32x$ .
Simplify the equation: Simplify the equation by canceling out the common factors.
Solve for $x$ : Solve for $x$ by isolating the variable.

Step-by-Step Solution

Let's apply the steps above to solve for $x$ using cross-products.

Step 1: Write the equation

The equation that we want to solve is $\frac{18}{x} = \frac{x}{32}$ .

Step 2: Multiply both sides by the LCM

To eliminate the fractions, we need to multiply both sides of the equation by the LCM of the denominators, which is $32x$ . This gives us:

32x \cdot \frac{18}{x} = 32x \cdot \frac{x}{32}

Step 3: Simplify the equation

Simplifying the equation, we get:

576 = x^2

Step 4: Solve for $x$

To solve for $x$ , we need to isolate the variable. Taking the square root of both sides, we get:

x = \pm \sqrt{576}

x = \pm 24

Conclusion

In this article, we have learned how to solve for $x$ using cross-products. We have applied the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ . We have also seen how to apply this technique to a specific equation, $\frac{18}{x} = \frac{x}{32}$ . By following these steps, we can solve for $x$ using cross-products and apply this technique to a wide range of equations.

Frequently Asked Questions

Q: What is cross-products?

A: Cross-products is a technique used to solve linear equations by multiplying both sides of the equation by a common factor.

Q: How do I apply cross-products to solve for $x$ ?

A: To apply cross-products, you need to follow the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ .

Q: What is the LCM of the denominators?

A: The LCM of the denominators is the least common multiple of the denominators of the fractions in the equation.

Example Problems

Problem 1

Solve for $x$ using cross-products:

\frac{24}{x} = \frac{x}{36}

Solution

To solve for $x$ , we need to follow the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ .

Write the equation: $\frac{24}{x} = \frac{x}{36}$
Multiply both sides by the LCM: $36x \cdot \frac{24}{x} = 36x \cdot \frac{x}{36}$
Simplify the equation: $864 = x^2$
Solve for $x$ : $x = \pm \sqrt{864}$

Problem 2

Solve for $x$ using cross-products:

\frac{48}{x} = \frac{x}{20}

Solution

To solve for $x$ , we need to follow the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ .

Write the equation: $\frac{48}{x} = \frac{x}{20}$
Multiply both sides by the LCM: $20x \cdot \frac{48}{x} = 20x \cdot \frac{x}{20}$
Simplify the equation: $960 = x^2$
Solve for $x$ : $x = \pm \sqrt{960}$

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Introduction

In our previous article, we explored how to solve for $x$ using cross-products. We learned how to apply this technique to a specific equation, $\frac{18}{x} = \frac{x}{32}$ , and saw how to follow the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ . In this article, we will continue to explore the concept of cross-products and answer some frequently asked questions.

Q&A

Q: What is the main advantage of using cross-products to solve for $x$ ?

A: The main advantage of using cross-products is that it allows us to eliminate the fractions in the equation, making it easier to solve for $x$ .

Q: How do I know when to use cross-products to solve for $x$ ?

A: You should use cross-products when you have a linear equation with fractions and you want to eliminate the fractions to solve for $x$ .

Q: Can I use cross-products to solve for $x$ in any type of equation?

A: No, you can only use cross-products to solve for $x$ in linear equations with fractions.

Q: What is the LCM of the denominators?

A: The LCM of the denominators is the least common multiple of the denominators of the fractions in the equation.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, you need to list the multiples of each denominator and find the smallest multiple that is common to both.

Q: Can I use cross-products to solve for $x$ in equations with variables on both sides?

A: Yes, you can use cross-products to solve for $x$ in equations with variables on both sides.

Q: How do I apply cross-products to solve for $x$ in equations with variables on both sides?

A: To apply cross-products to solve for $x$ in equations with variables on both sides, you need to follow the same steps as before: writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ .

Q: What is the difference between cross-products and other methods of solving for $x$ ?

A: The main difference between cross-products and other methods of solving for $x$ is that cross-products allows us to eliminate the fractions in the equation, making it easier to solve for $x$ .

Q: Can I use cross-products to solve for $x$ in equations with decimals?

A: Yes, you can use cross-products to solve for $x$ in equations with decimals.

Q: How do I apply cross-products to solve for $x$ in equations with decimals?

A: To apply cross-products to solve for $x$ in equations with decimals, you need to follow the same steps as before: writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ .

Example Problems

Problem 1

Solve for $x$ using cross-products:

\frac{24}{x} = \frac{x}{36}

Solution

To solve for $x$ , we need to follow the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ .

Write the equation: $\frac{24}{x} = \frac{x}{36}$
Multiply both sides by the LCM: $36x \cdot \frac{24}{x} = 36x \cdot \frac{x}{36}$
Simplify the equation: $864 = x^2$
Solve for $x$ : $x = \pm \sqrt{864}$

Problem 2

Solve for $x$ using cross-products:

\frac{48}{x} = \frac{x}{20}

Solution

To solve for $x$ , we need to follow the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ .

Write the equation: $\frac{48}{x} = \frac{x}{20}$
Multiply both sides by the LCM: $20x \cdot \frac{48}{x} = 20x \cdot \frac{x}{20}$
Simplify the equation: $960 = x^2$
Solve for $x$ : $x = \pm \sqrt{960}$

Conclusion

In this article, we have continued to explore the concept of cross-products and answered some frequently asked questions. We have seen how to apply cross-products to solve for $x$ in equations with variables on both sides and how to use cross-products to solve for $x$ in equations with decimals. By following the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ , we can use cross-products to solve for $x$ in a wide range of equations.

Frequently Asked Questions

Q: What is the main advantage of using cross-products to solve for $x$ ?

A: The main advantage of using cross-products is that it allows us to eliminate the fractions in the equation, making it easier to solve for $x$ .

Q: How do I know when to use cross-products to solve for $x$ ?

A: You should use cross-products when you have a linear equation with fractions and you want to eliminate the fractions to solve for $x$ .

Q: Can I use cross-products to solve for $x$ in any type of equation?

A: No, you can only use cross-products to solve for $x$ in linear equations with fractions.

Q: What is the LCM of the denominators?

A: The LCM of the denominators is the least common multiple of the denominators of the fractions in the equation.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, you need to list the multiples of each denominator and find the smallest multiple that is common to both.

Q: Can I use cross-products to solve for $x$ in equations with variables on both sides?

A: Yes, you can use cross-products to solve for $x$ in equations with variables on both sides.

Q: How do I apply cross-products to solve for $x$ in equations with variables on both sides?

Q: What is the difference between cross-products and other methods of solving for $x$ ?

A: The main difference between cross-products and other methods of solving for $x$ is that cross-products allows us to eliminate the fractions in the equation, making it easier to solve for $x$ .

Q: Can I use cross-products to solve for $x$ in equations with decimals?

A: Yes, you can use cross-products to solve for $x$ in equations with decimals.

Q: How do I apply cross-products to solve for $x$ in equations with decimals?

Example Problems

Problem 1

Solve for $x$ using cross-products:

\frac{24}{x} = \frac{x}{36}

Solution

To solve for $x$ , we need to follow the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ .

Write the equation: $\frac{24}{x} = \frac{x}{36}$
Multiply both sides by the LCM: $36x \cdot \frac{24}{x} = 36x \cdot \frac{x}{36}$
Simplify the equation: $864 = x^2$
Solve for $x$ : $x = \pm \sqrt{864}$

Problem 2

Solve for $x$ using cross-products:

\frac{48}{x} = \frac{x}{20}

Solution

To solve for $x$ , we need to follow the steps of writing the equation, multiplying both sides by the LCM, simplifying the equation, and solving for $x$ .

Write the equation: $\frac{48}{x} = \frac{x}{20}$
Multiply both sides by the LCM: $20x \cdot \frac{48}{x} = 20x \cdot \frac{x}{20}$
Simplify the equation: $960 = x^2$
Solve for $x$ : $x = \pm \sqrt{960}$