Solve For $x$:$3 - 2(2x - 3) + 3(4x - 1) = 20$

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of linear equation, which involves simplifying and isolating the variable x. We will use the given equation $3 - 2(2x - 3) + 3(4x - 1) = 20$ as an example to demonstrate the step-by-step process of solving for x.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, x) is 1. The equation is also a combination of addition, subtraction, multiplication, and parentheses, which makes it a bit more complex than a simple linear equation.

Step 1: Distribute the Numbers Outside the Parentheses

The first step in solving the equation is to distribute the numbers outside the parentheses to the terms inside. This means multiplying the number outside the parentheses by each term inside the parentheses.

3−2(2x−3)+3(4x−1)=203 - 2(2x - 3) + 3(4x - 1) = 20

Distributing the numbers outside the parentheses, we get:

3−4x+6+12x−3=203 - 4x + 6 + 12x - 3 = 20

Step 2: Combine Like Terms

The next step is to combine like terms, which means combining terms that have the same variable and exponent. In this case, we can combine the constant terms (3 and 6) and the x terms (-4x and 12x).

3−4x+6+12x−3=203 - 4x + 6 + 12x - 3 = 20

Combining like terms, we get:

6+8x−3=206 + 8x - 3 = 20

Step 3: Simplify the Equation

Now that we have combined like terms, we can simplify the equation by combining the constant terms.

6+8x−3=206 + 8x - 3 = 20

Simplifying the equation, we get:

3+8x=203 + 8x = 20

Step 4: Isolate the Variable x

The final step is to isolate the variable x by getting rid of the constant term on the left side of the equation. We can do this by subtracting 3 from both sides of the equation.

3+8x=203 + 8x = 20

Subtracting 3 from both sides, we get:

8x=178x = 17

Step 5: Solve for x

Now that we have isolated the variable x, we can solve for x by dividing both sides of the equation by 8.

8x=178x = 17

Dividing both sides by 8, we get:

x=178x = \frac{17}{8}

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the step-by-step process outlined in this article, we can solve even the most complex linear equations. Remember to distribute the numbers outside the parentheses, combine like terms, simplify the equation, isolate the variable x, and finally solve for x. With practice and patience, you will become a master of solving linear equations in no time.

Example Use Cases

Solving linear equations has numerous real-world applications, including:

  • Physics and Engineering: Linear equations are used to describe the motion of objects, the flow of fluids, and the behavior of electrical circuits.
  • Economics: Linear equations are used to model the behavior of economic systems, including supply and demand curves.
  • Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations like a pro:

  • Use the distributive property: The distributive property states that a(b + c) = ab + ac. Use this property to simplify complex equations.
  • Combine like terms: Combine terms that have the same variable and exponent to simplify the equation.
  • Simplify the equation: Simplify the equation by combining constant terms and eliminating parentheses.
  • Isolate the variable x: Get rid of the constant term on the left side of the equation by subtracting or adding it to both sides.
  • Practice, practice, practice: The more you practice solving linear equations, the more comfortable you will become with the process.

Conclusion

Introduction

Solving linear equations is a crucial skill for students and professionals alike. In our previous article, we provided a step-by-step guide on how to solve linear equations. However, we understand that sometimes, it's easier to learn through questions and answers. In this article, we will provide a Q&A guide on solving linear equations, covering common questions and scenarios.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. It is a simple equation that can be solved by following a few basic steps.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Distribute the numbers outside the parentheses to the terms inside.
  2. Combine like terms.
  3. Simplify the equation.
  4. Isolate the variable x.
  5. Solve for x.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that a(b + c) = ab + ac. It is used to simplify complex equations by distributing the numbers outside the parentheses to the terms inside.

Q: How do I combine like terms?

A: To combine like terms, you need to identify the terms that have the same variable and exponent. Then, you can add or subtract the coefficients of those terms to simplify the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable (in this case, x) is 2. Quadratic equations are more complex and require different techniques to solve.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to understand the steps involved in solving the equation, so you can verify the answer and learn from the process.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

  • Not distributing the numbers outside the parentheses to the terms inside.
  • Not combining like terms.
  • Not simplifying the equation.
  • Not isolating the variable x.
  • Not solving for x.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by:

  • Working on sample problems and exercises.
  • Using online resources and calculators.
  • Joining a study group or seeking help from a tutor.
  • Creating your own problems and solving them.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications, including:

  • Physics and Engineering: Linear equations are used to describe the motion of objects, the flow of fluids, and the behavior of electrical circuits.
  • Economics: Linear equations are used to model the behavior of economic systems, including supply and demand curves.
  • Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article and practicing regularly, you can become proficient in solving linear equations. Remember to distribute the numbers outside the parentheses, combine like terms, simplify the equation, isolate the variable x, and finally solve for x. With practice and patience, you will become a master of solving linear equations in no time.