Type The Correct Answer In The Box. Use Numerals Instead Of Words.What Value Of $n$ Makes The Equation True? − 1 5 N + 7 = 2 -\frac{1}{5}n + 7 = 2 − 5 1 ​ N + 7 = 2 N = N = N = □ \square □

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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 linear equations with one variable, specifically the equation −15n+7=2n-\frac{1}{5}n + 7 = 2n. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.

Understanding the Equation

The given equation is −15n+7=2n-\frac{1}{5}n + 7 = 2n. To solve for nn, we need to isolate the variable nn on one side of the equation. The equation has two terms involving nn: −15n-\frac{1}{5}n and 2n2n. Our goal is to get rid of the fraction and combine like terms.

Step 1: Get Rid of the Fraction

To eliminate the fraction, we can multiply both sides of the equation by the denominator of the fraction, which is 5. This will give us:

5×(−15n+7)=5×2n5 \times \left(-\frac{1}{5}n + 7\right) = 5 \times 2n

Simplifying the left-hand side, we get:

−n+35=10n-n + 35 = 10n

Step 2: Combine Like Terms

Now, we can combine the like terms on the left-hand side of the equation. We have −n-n and 3535, which are both constants. We can add them together to get:

35=11n35 = 11n

Step 3: Solve for nn

To solve for nn, we need to isolate the variable nn on one side of the equation. We can do this by dividing both sides of the equation by 11:

3511=n\frac{35}{11} = n

Conclusion

In this article, we solved the linear equation −15n+7=2n-\frac{1}{5}n + 7 = 2n step by step. We eliminated the fraction, combined like terms, and solved for the variable nn. The final answer is 3511\boxed{\frac{35}{11}}.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
  • To eliminate fractions, multiply both sides of the equation by the denominator of the fraction.
  • Combine like terms by adding or subtracting constants.
  • To solve for a variable, isolate it on one side of the equation by performing inverse operations.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Finance: Linear equations are used to calculate interest rates, investments, and loans.
  • Science: Linear equations are used to model population growth, chemical reactions, and physical systems.
  • Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Practice Problems

Try solving the following linear equations:

  1. 2x+5=112x + 5 = 11
  2. x−3=7x - 3 = 7
  3. 4y−2=104y - 2 = 10

Conclusion

Solving linear equations is a fundamental skill that has numerous real-world applications. By following the steps outlined in this article, you can master the art of solving linear equations and apply it to various fields. Remember to eliminate fractions, combine like terms, and solve for the variable. With practice, you'll become proficient in solving linear equations and tackle more complex problems with confidence.

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Introduction

Solving linear equations can be a challenging task, especially for students who are new to algebra. In this article, we will address some of the most frequently asked questions about solving linear equations. Whether you're a student, teacher, or simply someone who wants to brush up on their math skills, this article is for you.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x or y) is 1. In other words, it's an equation that 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 isolate the variable (usually x or y) on one side of the equation. You can do this by performing inverse operations, such as addition, subtraction, multiplication, and division.

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:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I eliminate fractions in a linear equation?

A: To eliminate fractions in a linear equation, you can multiply both sides of the equation by the denominator of the fraction. This will get rid of the fraction and allow you 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, the equation x + 2 = 3 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.

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 check your work by hand to make sure you understand the solution.

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 eliminating fractions
  • Not combining like terms
  • Not checking your work

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

  • Working through practice problems in a textbook or online resource
  • Using a calculator to solve equations and then checking your work by hand
  • Creating your own practice problems and solving them
  • Joining a study group or working with a tutor to practice solving linear equations

Conclusion

Solving linear equations is a fundamental skill that has numerous real-world applications. By following the steps outlined in this article and practicing regularly, you can become proficient in solving linear equations and tackle more complex problems with confidence. Remember to eliminate fractions, combine like terms, and solve for the variable. With practice, you'll become a master of solving linear equations!