Solve For { N $} . . . { 7n + 2 = 4n + 17 \}
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 is a first-degree equation in one variable. We will use the given equation 7n + 2 = 4n + 17 as an example to demonstrate the step-by-step process of solving linear equations.
What are Linear Equations?
A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.
The Given Equation
The given equation is 7n + 2 = 4n + 17. This equation is a linear equation in one variable, n. Our goal is to solve for n, which means we need to isolate the variable n on one side of the equation.
Step 1: Subtract 4n from Both Sides
To solve for n, we need to get all the terms involving n on one side of the equation. We can do this by subtracting 4n from both sides of the equation. This will give us:
7n - 4n + 2 = 4n - 4n + 17
Simplifying the equation, we get:
3n + 2 = 17
Step 2: Subtract 2 from Both Sides
Next, we need to get rid of the constant term on the left-hand side of the equation. We can do this by subtracting 2 from both sides of the equation. This will give us:
3n + 2 - 2 = 17 - 2
Simplifying the equation, we get:
3n = 15
Step 3: Divide Both Sides by 3
Finally, we need to isolate the variable n by dividing both sides of the equation by 3. This will give us:
3n / 3 = 15 / 3
Simplifying the equation, we get:
n = 5
Conclusion
In this article, we solved the linear equation 7n + 2 = 4n + 17 using the step-by-step process of subtracting 4n from both sides, subtracting 2 from both sides, and dividing both sides by 3. We found that the value of n is 5. This demonstrates the importance of following the correct steps when solving linear equations.
Tips and Tricks
Here are some tips and tricks to help you solve linear equations:
- Use inverse operations: When solving linear equations, use inverse operations to isolate the variable. For example, if you have a term with a coefficient of 2, you can use the inverse operation of division to get rid of the coefficient.
- Simplify the equation: Simplify the equation as much as possible to make it easier to solve.
- Check your work: Always check your work by plugging the solution back into the original equation.
Real-World Applications
Linear equations have many real-world applications, including:
- Physics and engineering: Linear equations are used to describe the motion of objects and the behavior of physical systems.
- Economics: Linear equations are used to model economic systems and make predictions about economic trends.
- Computer science: Linear equations are used in computer algorithms and data analysis.
Common Mistakes
Here are some common mistakes to avoid when solving linear equations:
- Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving linear equations.
- Not simplifying the equation: Simplify the equation as much as possible to make it easier to solve.
- Not checking your work: Always check your work by plugging the solution back into the original equation.
Conclusion
Introduction
In our previous article, we discussed the step-by-step process of solving linear equations. However, we understand that sometimes, it's not enough to just provide a solution. That's why we've put together this Q&A guide to help you better understand the concept of solving linear equations.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by using inverse operations, such as addition, subtraction, multiplication, and division.
Q: What are inverse operations?
A: Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division.
Q: How do I use inverse operations to solve a linear equation?
A: To use inverse operations to solve a linear equation, you need to identify the operation that is being performed on the variable. Then, you need to perform the inverse operation to isolate the variable.
Q: What is the order of operations?
A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I simplify an equation?
A: To simplify an equation, you need to combine like terms and eliminate any unnecessary operations.
Q: What are like terms?
A: Like terms are terms that have the same variable and coefficient. For example, 2x and 4x are like terms, as are 3y and 5y.
Q: How do I check my work?
A: To check your work, you need to plug your solution back into the original equation and make sure it is true.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not following the order of operations
- Not simplifying the equation
- Not checking your work
- Not using inverse operations to isolate the variable
Q: How do I apply linear equations to real-world problems?
A: Linear equations can be applied to a wide range of real-world problems, including:
- Physics and engineering: Linear equations are used to describe the motion of objects and the behavior of physical systems.
- Economics: Linear equations are used to model economic systems and make predictions about economic trends.
- Computer science: Linear equations are used in computer algorithms and data analysis.
Conclusion
In conclusion, solving linear equations is an essential skill for students and professionals alike. By following the step-by-step process of using inverse operations, simplifying the equation, and checking your work, you can solve linear equations and apply them to real-world problems. Remember to avoid common mistakes and use the order of operations to ensure accuracy.
Additional Resources
If you're looking for additional resources to help you learn more about solving linear equations, here are a few suggestions:
- Khan Academy: Khan Academy has a comprehensive video series on solving linear equations.
- Mathway: Mathway is an online math problem solver that can help you solve linear equations and other math problems.
- Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve linear equations and other math problems.
Final Tips
Here are a few final tips to help you master the art of solving linear equations:
- Practice, practice, practice: The more you practice solving linear equations, the more comfortable you'll become with the process.
- Use online resources: There are many online resources available to help you learn more about solving linear equations, including video tutorials and online calculators.
- Seek help when needed: Don't be afraid to ask for help if you're struggling with a particular problem or concept.