Solve The Equation For { Q $} : : : { 2(2q + 1.5) = 18 - Q \}

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation involving a variable, q. The equation is given as: 2(2q + 1.5) = 18 - q. We will break down the solution step by step, using a clear and concise approach.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, q) is 1. The equation is also a quadratic equation in disguise, as it can be rewritten as a quadratic equation by multiplying out the parentheses.

Step 1: Distribute the 2

To solve the equation, we need to start by distributing the 2 to the terms inside the parentheses. This will give us: 4q + 3 = 18 - q.

Step 2: Add q to Both Sides

Next, we need to add q to both sides of the equation to get all the q terms on one side. This will give us: 4q + q + 3 = 18.

Step 3: Combine Like Terms

Now, we can combine the like terms on the left-hand side of the equation. This will give us: 5q + 3 = 18.

Step 4: Subtract 3 from Both Sides

Next, we need to subtract 3 from both sides of the equation to get rid of the constant term on the left-hand side. This will give us: 5q = 15.

Step 5: Divide Both Sides by 5

Finally, we need to divide both sides of the equation by 5 to solve for q. This will give us: q = 3.

Conclusion

In this article, we have solved a linear equation involving a variable, q. We have broken down the solution into five steps, using a clear and concise approach. By following these steps, we have arrived at the solution: q = 3. This is a fundamental concept in mathematics, and solving linear equations is a crucial skill for students to master.

Tips and Tricks

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

  • Always start by distributing any coefficients to the terms inside the parentheses.
  • Add or subtract the same value to both sides of the equation to get all the terms with the variable on one side.
  • Combine like terms on the left-hand side of the equation.
  • Subtract a constant term from both sides of the equation to get rid of it.
  • Divide both sides of the equation by a coefficient to solve for the variable.

Real-World Applications

Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

  • Not distributing coefficients to terms inside parentheses.
  • Not adding or subtracting the same value to both sides of the equation.
  • Not combining like terms on the left-hand side of the equation.
  • Not subtracting a constant term from both sides of the equation.
  • Not dividing both sides of the equation by a coefficient to solve for the variable.

Conclusion

Introduction

In our previous article, we covered the basics of solving linear equations. However, we know that practice makes perfect, and sometimes, it's helpful to have a Q&A guide to clarify any doubts. In this article, we will answer some frequently asked questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, q) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to follow these steps:

  1. Distribute any coefficients to the terms inside the parentheses.
  2. Add or subtract the same value to both sides of the equation to get all the terms with the variable on one side.
  3. Combine like terms on the left-hand side of the equation.
  4. Subtract a constant term from both sides of the equation to get rid of it.
  5. Divide both sides of the equation by a coefficient to solve for the variable.

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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, 2x + 3 = 5 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, you need to look at the highest power of the variable. If the highest power is 1, it's a linear equation. If the highest power is 2, it's a quadratic equation.

Q: Can I use algebraic methods to solve linear equations?

A: Yes, you can use algebraic methods to solve linear equations. In fact, algebraic methods are often the most efficient way to solve linear equations.

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 coefficients to terms inside parentheses.
  • Not adding or subtracting the same value to both sides of the equation.
  • Not combining like terms on the left-hand side of the equation.
  • Not subtracting a constant term from both sides of the equation.
  • Not dividing both sides of the equation by a coefficient to solve for the variable.

Q: Can I use technology to solve linear equations?

A: Yes, you can use technology to solve linear equations. Many calculators and computer software programs can solve linear equations quickly and accurately.

Q: How do I check my work when solving linear equations?

A: To check your work when solving linear equations, you need to plug your solution back into the original equation and make sure it's true. This will help you ensure that your solution is correct.

Conclusion

In conclusion, solving linear equations is a fundamental concept in mathematics, and it has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving linear equations and apply them to real-world problems. Remember to always distribute coefficients, add or subtract the same value to both sides of the equation, combine like terms, subtract a constant term, and divide both sides of the equation by a coefficient to solve for the variable. With practice and patience, you can become a master of solving linear equations.