Solving 3q + 5 + 2q - 5 = 65 A Step-by-Step Guide

Introduction

In the realm of mathematics, solving equations is a fundamental skill. It's the bedrock upon which more advanced concepts are built. Among the various types of equations, linear equations hold a special place due to their simplicity and wide applicability. This article will delve into the process of simplifying and solving a linear equation, providing a step-by-step guide to help you master this essential skill. We'll specifically address the equation 3q + 5 + 2q - 5 = 65, breaking down each step to ensure clarity and understanding.

Understanding Linear Equations

Before we dive into the solution, let's first understand what a linear equation is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable's exponent is always 1, and the equation, when graphed, forms a straight line. Linear equations are ubiquitous in mathematics and various real-world applications, from calculating distances and speeds to modeling financial growth and decay. They form the basis for more complex algebraic structures and are crucial for problem-solving in numerous fields.

The key characteristics of a linear equation include:

• Single Variable: Typically, linear equations involve only one variable (in our case, 'q').
• Exponent of 1: The variable is raised to the power of 1.
• Constants: The equation includes constant terms (numbers) that do not change.
• Equality: The equation has an equality sign (=), indicating that the expressions on both sides have the same value.

The Importance of Simplifying Equations

Simplifying an equation before solving it is akin to clearing a path before embarking on a journey. It makes the problem more manageable and reduces the chances of making errors. A simplified equation is easier to solve because it has fewer terms and operations, making the steps to isolate the variable more straightforward. Simplification involves combining like terms, which are terms that contain the same variable raised to the same power, and eliminating constants where possible.

In our equation, 3q + 5 + 2q - 5 = 65, simplification involves combining the 'q' terms (3q and 2q) and the constant terms (5 and -5). This process not only makes the equation less cluttered but also highlights the underlying structure, making the subsequent steps in the solution clearer. Simplifying equations is a skill that extends beyond linear equations, proving valuable in solving quadratic, polynomial, and even trigonometric equations.

Step-by-Step Solution: Simplifying and Solving 3q + 5 + 2q - 5 = 65

Step 1: Combine Like Terms

This is the first critical step in simplifying the equation. Combine like terms on the left side of the equation. Like terms are terms that contain the same variable raised to the same power. In our equation, 3q and 2q are like terms, and +5 and -5 are like terms. Combining like terms involves adding or subtracting their coefficients (the numbers in front of the variable) or the constants themselves.

• Combine the 'q' terms: 3q + 2q = 5q
• Combine the constant terms: +5 - 5 = 0

After combining like terms, the equation becomes:

• 5q + 0 = 65

Which simplifies further to:

• 5q = 65

This step is crucial because it reduces the complexity of the equation, making it easier to solve. By grouping similar terms, we streamline the equation and bring it closer to a form where we can isolate the variable 'q'.

Step 2: Isolate the Variable

Now that we have simplified the equation to 5q = 65, the next step is to isolate the variable 'q'. This means getting 'q' by itself on one side of the equation. In this case, 'q' is being multiplied by 5. To isolate 'q', we need to perform the inverse operation, which is division. The golden rule of equation solving is that whatever operation you perform on one side of the equation, you must perform on the other side to maintain equality.

Divide both sides of the equation by 5:

• (5q) / 5 = 65 / 5

This simplifies to:

• q = 13

By dividing both sides by 5, we effectively "undo" the multiplication, leaving 'q' isolated on the left side. This step is the heart of solving for 'q', as it reveals the value of the variable that satisfies the equation.

Step 3: Verify the Solution

After finding a potential solution, it's always a good practice to verify the solution. This involves substituting the value you found for 'q' back into the original equation to check if it makes the equation true. This step is a safeguard against errors, ensuring that you have indeed found the correct value of 'q'.

Substitute q = 13 into the original equation 3q + 5 + 2q - 5 = 65:

• 3(13) + 5 + 2(13) - 5 = 65

Perform the multiplication:

• 39 + 5 + 26 - 5 = 65

Perform the addition and subtraction:

• 65 = 65

Since the equation holds true, q = 13 is indeed the correct solution. This verification step confirms our solution and reinforces our confidence in the process.

Common Mistakes to Avoid

Solving equations can sometimes be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid when simplifying and solving linear equations:

• Incorrectly Combining Like Terms: Ensure you only combine terms with the same variable and exponent. For instance, 3q and 2q can be combined, but 3q and 2q² cannot.
• Forgetting the Distributive Property: If there are parentheses in the equation, remember to distribute any factors outside the parentheses to each term inside.
• Incorrectly Applying Inverse Operations: To isolate the variable, you must use the correct inverse operation. If the variable is being multiplied, divide; if it's being added, subtract.
• Not Performing the Same Operation on Both Sides: Always maintain equality by performing the same operation on both sides of the equation.
• Skipping the Verification Step: It's crucial to verify your solution by substituting it back into the original equation. This helps catch any errors made during the solving process.
• Sign Errors: Pay close attention to the signs (+ and -) of the terms. A simple sign error can lead to an incorrect solution.

By being aware of these common pitfalls and taking your time to work through each step carefully, you can minimize the chances of making mistakes and improve your equation-solving skills.

Real-World Applications of Linear Equations

Linear equations are not just abstract mathematical concepts; they have a plethora of real-world applications. They are used to model and solve problems in various fields, including:

• Finance: Calculating simple and compound interest, determining loan payments, and analyzing investments.
• Physics: Describing motion with constant velocity, analyzing forces, and calculating electrical circuits.
• Engineering: Designing structures, optimizing processes, and modeling systems.
• Economics: Predicting supply and demand, analyzing market trends, and calculating economic growth.
• Everyday Life: Calculating distances and travel times, determining costs and budgets, and making informed decisions.

For example, consider a scenario where you want to determine how long it will take to drive a certain distance at a constant speed. This can be modeled using a linear equation where distance equals rate multiplied by time (d = rt). Similarly, calculating the total cost of a taxi ride, which includes a fixed charge plus a per-mile fee, can also be represented by a linear equation.

The versatility of linear equations makes them an indispensable tool for problem-solving in a wide range of contexts. Understanding how to simplify and solve them is not just an academic exercise but a practical skill that can be applied in numerous aspects of life.

Conclusion

Mastering the art of simplifying and solving linear equations is a fundamental step in your mathematical journey. By following the steps outlined in this article – combining like terms, isolating the variable, and verifying the solution – you can confidently tackle a wide range of linear equations. Remember to avoid common mistakes, such as incorrectly combining like terms or neglecting to perform the same operation on both sides of the equation. The solution to the equation 3q + 5 + 2q - 5 = 65 is q = 13, which we verified by substituting it back into the original equation.

Linear equations are not confined to the classroom; they are powerful tools that have real-world applications in various fields, from finance and physics to engineering and everyday life. The ability to solve them empowers you to analyze and solve problems in a systematic and logical manner. As you continue your mathematical studies, the skills you've learned here will serve as a solid foundation for more advanced concepts and techniques. Embrace the challenge of solving equations, and you'll find yourself equipped with a valuable problem-solving skill that will benefit you throughout your academic and professional endeavors.

Practice Problems

To solidify your understanding of simplifying and solving linear equations, here are some practice problems for you to try:

1. Solve for x: 2x + 7 - x + 3 = 15
2. Solve for y: 4y - 9 - y + 1 = 10
3. Solve for z: 5z + 2 + 3z - 8 = 20
4. Solve for a: 6a - 4 + 2a + 10 = 30
5. Solve for b: 7b + 1 - b - 5 = 14

Work through these problems step-by-step, applying the techniques discussed in this article. Remember to simplify the equations first by combining like terms, then isolate the variable, and finally, verify your solution. The more you practice, the more confident and proficient you will become in solving linear equations.