Solve The Equation:${ \begin{array}{l} 10(2 \omega - 5) = 2 \omega + 2(2 \omega + 1) \ 20 \omega - 50 = 2 \omega + 4 \omega + 2 \end{array} }$

Introduction

In this article, we will delve into the world of mathematics and solve a complex equation step by step. The equation in question is a linear equation that involves a variable, ω. Our goal is to isolate the variable and find its value. We will use algebraic techniques to simplify the equation and solve for ω.

The Equation

The equation we need to solve is:

$$\begin{array}{l} 10(2 \omega - 5) = 2 \omega + 2(2 \omega + 1) \\ 20 \omega - 50 = 2 \omega + 4 \omega + 2 \end{array}$$

Step 1: Simplify the Left Side of the Equation

To simplify the left side of the equation, we need to distribute the 10 to the terms inside the parentheses.

$$10(2 \omega - 5) = 20 \omega - 50$$

Step 2: Simplify the Right Side of the Equation

To simplify the right side of the equation, we need to distribute the 2 to the terms inside the parentheses.

$$2(2 \omega + 1) = 4 \omega + 2$$

Step 3: Rewrite the Equation

Now that we have simplified both sides of the equation, we can rewrite it as:

$$20 \omega - 50 = 2 \omega + 4 \omega + 2$$

Step 4: Combine Like Terms

To combine like terms, we need to group the terms with the same variable (ω) together.

$$20 \omega - 50 = 6 \omega + 2$$

Step 5: Isolate the Variable

To isolate the variable, we need to get all the terms with ω on one side of the equation. We can do this by subtracting 6ω from both sides of the equation.

$$20 \omega - 6 \omega - 50 = 6 \omega - 6 \omega + 2$$

Step 6: Simplify the Equation

Now that we have isolated the variable, we can simplify the equation by combining like terms.

$$14 \omega - 50 = 2$$

Step 7: Add 50 to Both Sides

To get rid of the negative term, we can add 50 to both sides of the equation.

$$14 \omega - 50 + 50 = 2 + 50$$

Step 8: Simplify the Equation

Now that we have added 50 to both sides of the equation, we can simplify it by combining like terms.

$$14 \omega = 52$$

Step 9: Divide Both Sides by 14

To find the value of ω, we need to divide both sides of the equation by 14.

$$\frac{14 \omega}{14} = \frac{52}{14}$$

Step 10: Simplify the Equation

Now that we have divided both sides of the equation by 14, we can simplify it by combining like terms.

$$\omega = \frac{52}{14}$$

Step 11: Simplify the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of 52 and 14.

$$\text{GCD}(52, 14) = 2$$

Step 12: Divide Both Numerator and Denominator by the GCD

To simplify the fraction, we need to divide both the numerator and the denominator by the GCD.

$$\omega = \frac{\frac{52}{2}}{\frac{14}{2}}$$

Step 13: Simplify the Fraction

Now that we have divided both the numerator and the denominator by the GCD, we can simplify the fraction.

$$\omega = \frac{26}{7}$$

Conclusion

In this article, we have solved a complex equation step by step. We have used algebraic techniques to simplify the equation and find the value of the variable ω. The final answer is:

$$\omega = \frac{26}{7}$$

Discussion

The equation we solved is a linear equation that involves a variable, ω. We used algebraic techniques to simplify the equation and find the value of ω. The final answer is a fraction, which can be simplified further by dividing both the numerator and the denominator by their greatest common divisor.

Related Topics

• Linear equations
• Algebraic techniques
• Simplifying fractions

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra" by Jim Hefferon

Keywords

• Linear equations
• Algebraic techniques
• Simplifying fractions
• Variable ω
• Greatest common divisor (GCD)
  Solving the Equation: A Q&A Guide =====================================

Introduction

In our previous article, we solved a complex equation step by step. In this article, we will answer some frequently asked questions (FAQs) related to the equation and its solution.

Q: What is the equation we solved?

A: The equation we solved is:

$$\begin{array}{l} 10(2 \omega - 5) = 2 \omega + 2(2 \omega + 1) \\ 20 \omega - 50 = 2 \omega + 4 \omega + 2 \end{array}$$

Q: What is the value of ω?

A: The value of ω is:

$$\omega = \frac{26}{7}$$

Q: How did we simplify the equation?

A: We simplified the equation by using algebraic techniques, such as distributing the 10 to the terms inside the parentheses, combining like terms, and isolating the variable.

Q: What is the greatest common divisor (GCD) of 52 and 14?

A: The greatest common divisor (GCD) of 52 and 14 is 2.

Q: Why did we divide both the numerator and the denominator by the GCD?

A: We divided both the numerator and the denominator by the GCD to simplify the fraction.

Q: What is the final answer?

A: The final answer is:

$$\omega = \frac{26}{7}$$

Q: Can you explain the steps in more detail?

A: Of course! Here are the steps in more detail:

1. Simplify the left side of the equation by distributing the 10 to the terms inside the parentheses.
2. Simplify the right side of the equation by distributing the 2 to the terms inside the parentheses.
3. Rewrite the equation with the simplified left and right sides.
4. Combine like terms by grouping the terms with the same variable (ω) together.
5. Isolate the variable by getting all the terms with ω on one side of the equation.
6. Simplify the equation by combining like terms.
7. Add 50 to both sides of the equation to get rid of the negative term.
8. Simplify the equation by combining like terms.
9. Divide both sides of the equation by 14 to find the value of ω.
10. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Q: What are some related topics?

A: Some related topics include:

• Linear equations
• Algebraic techniques
• Simplifying fractions
• Variable ω
• Greatest common divisor (GCD)

Q: Where can I find more information?

A: You can find more information in the following resources:

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra" by Jim Hefferon

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to the equation and its solution. We hope this article has been helpful in clarifying any doubts you may have had.

Discussion

The equation we solved is a linear equation that involves a variable, ω. We used algebraic techniques to simplify the equation and find the value of ω. The final answer is a fraction, which can be simplified further by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Related Topics

• Linear equations
• Algebraic techniques
• Simplifying fractions
• Variable ω
• Greatest common divisor (GCD)

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra" by Jim Hefferon

Keywords

• Linear equations
• Algebraic techniques
• Simplifying fractions
• Variable ω
• Greatest common divisor (GCD)