Add or subtract incorrectly

Oct 13, 2025 by magicalceylontours.com 145 views

Introduction

Rational equations are a fundamental concept in mathematics, and solving for x is a crucial step in understanding these equations. In this article, we will explore how to solve for x in a rational equation, using the given equation as an example.

The Given Equation

The given equation is:

\frac{x-3}{18} = \frac{12}{9}

Step 1: Cross-Multiply

To solve for x, we need to eliminate the fractions. We can do this by cross-multiplying, which involves multiplying both sides of the equation by the denominators of the fractions.

\frac{x-3}{18} = \frac{12}{9}

\Rightarrow (x-3) \times 9 = 18 \times 12

Step 2: Simplify

Now, we can simplify the equation by multiplying the numbers.

\Rightarrow 9x - 27 = 216

Step 3: Add 27 to Both Sides

To isolate the term with x, we need to add 27 to both sides of the equation.

\Rightarrow 9x - 27 + 27 = 216 + 27

\Rightarrow 9x = 243

Step 4: Divide by 9

Finally, we can solve for x by dividing both sides of the equation by 9.

\Rightarrow \frac{9x}{9} = \frac{243}{9}

\Rightarrow x = 27

Conclusion

In this article, we have solved for x in a rational equation using the given equation as an example. We have used cross-multiplication, simplification, and division to isolate the term with x. The final answer is x = 27.

Example Use Cases

Rational equations are used in a variety of real-world applications, including:

Finance: Rational equations are used to calculate interest rates and investment returns.
Science: Rational equations are used to model population growth and chemical reactions.
Engineering: Rational equations are used to design and optimize systems, such as electrical circuits and mechanical systems.

Tips and Tricks

When solving for x in a rational equation, remember to:

Cross-multiply: Eliminate the fractions by multiplying both sides of the equation by the denominators.
Simplify: Multiply the numbers and combine like terms.
Add or subtract: Isolate the term with x by adding or subtracting the same value from both sides.
Divide: Solve for x by dividing both sides of the equation by the coefficient of x.

Common Mistakes

When solving for x in a rational equation, be careful not to:

Forget to cross-multiply: Failing to eliminate the fractions can lead to incorrect solutions.
Simplify incorrectly: Misinterpreting the signs or values of the numbers can lead to incorrect solutions.
Add or subtract incorrectly: Failing to add or subtract the same value from both sides can lead to incorrect solutions.
Divide incorrectly: Failing to divide both sides of the equation by the coefficient of x can lead to incorrect solutions.

Conclusion

Introduction

In our previous article, we explored how to solve for x in a rational equation. In this article, we will answer some common questions and provide additional examples to help you master this concept.

Q: What is a rational equation?

A: A rational equation is an equation that contains fractions, where the numerator and denominator are polynomials.

Q: Why do we need to solve for x in a rational equation?

A: Solving for x in a rational equation allows us to find the value of the variable x, which is essential in many real-world applications, such as finance, science, and engineering.

Q: What are the steps to solve for x in a rational equation?

A: The steps to solve for x in a rational equation are:

Cross-multiply: Eliminate the fractions by multiplying both sides of the equation by the denominators.
Simplify: Multiply the numbers and combine like terms.
Add or subtract: Isolate the term with x by adding or subtracting the same value from both sides.
Divide: Solve for x by dividing both sides of the equation by the coefficient of x.

Q: What are some common mistakes to avoid when solving for x in a rational equation?

A: Some common mistakes to avoid when solving for x in a rational equation include:

Forgetting to cross-multiply: Failing to eliminate the fractions can lead to incorrect solutions.
Simplifying incorrectly: Misinterpreting the signs or values of the numbers can lead to incorrect solutions.
Adding or subtracting incorrectly: Failing to add or subtract the same value from both sides can lead to incorrect solutions.
Dividing incorrectly: Failing to divide both sides of the equation by the coefficient of x can lead to incorrect solutions.

Q: How do I know if I have solved for x correctly?

A: To ensure that you have solved for x correctly, follow these steps:

Check your work: Verify that you have followed the steps correctly and that your solution is consistent with the original equation.
Plug in your solution: Substitute your solution back into the original equation to verify that it is true.
Check for extraneous solutions: Verify that your solution is not an extraneous solution, which is a solution that is not valid in the context of the problem.

Q: What are some real-world applications of solving for x in a rational equation?

A: Solving for x in a rational equation has many real-world applications, including:

Finance: Rational equations are used to calculate interest rates and investment returns.
Science: Rational equations are used to model population growth and chemical reactions.
Engineering: Rational equations are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: Can you provide some additional examples of solving for x in a rational equation?

A: Here are some additional examples of solving for x in a rational equation:

Example 1:

\frac{x+2}{4} = \frac{3}{2}

Solution:

Cross-multiply: $\frac{x+2}{4} = \frac{3}{2} \Rightarrow (x+2) \times 2 = 4 \times 3$
Simplify: $2x + 4 = 12$
Add or subtract: $2x + 4 - 4 = 12 - 4 \Rightarrow 2x = 8$
Divide: $\frac{2x}{2} = \frac{8}{2} \Rightarrow x = 4$

Example 2:

\frac{x-1}{3} = \frac{2}{5}

Solution:

Cross-multiply: $\frac{x-1}{3} = \frac{2}{5} \Rightarrow (x-1) \times 5 = 3 \times 2$
Simplify: $5x - 5 = 6$
Add or subtract: $5x - 5 + 5 = 6 + 5 \Rightarrow 5x = 11$
Divide: $\frac{5x}{5} = \frac{11}{5} \Rightarrow x = \frac{11}{5}$

Conclusion

Solving for x in a rational equation is a crucial step in understanding these equations. By following the steps outlined in this article, you can solve for x in a rational equation and apply this knowledge to real-world applications. Remember to cross-multiply, simplify, add or subtract, and divide to isolate the term with x.