Solve This Equation For $k$. 3 − 3 K + 7 K = 5 B 3 - 3k + 7k = 5b 3 − 3 K + 7 K = 5 B Enter The Correct Answer In The Box. K = □ □ K = \frac{\square}{\square} K = □ □ ​

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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, $3 - 3k + 7k = 5b$, to find the value of the variable $k$. We will break down the solution step by step, using algebraic techniques to isolate the variable and find its value.

Understanding the Equation

Before we start solving the equation, let's take a closer look at its structure. The equation is a linear equation, which means it is in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this case, the equation is $3 - 3k + 7k = 5b$.

Step 1: Combine Like Terms

The first step in solving the equation is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $k$, $-3k$ and $7k$. We can combine these terms by adding their coefficients.

3−3k+7k=3+(7k−3k)3 - 3k + 7k = 3 + (7k - 3k)

Simplifying the expression, we get:

3+4k3 + 4k

So, the equation becomes:

3+4k=5b3 + 4k = 5b

Step 2: Isolate the Variable

Now that we have combined like terms, we need to isolate the variable $k$. To do this, we need to get all the terms with $k$ on one side of the equation and the constant terms on the other side.

We can start by subtracting $3$ from both sides of the equation:

4k=5b−34k = 5b - 3

Step 3: Solve for k

Now that we have isolated the variable $k$, we can solve for its value. To do this, we need to divide both sides of the equation by the coefficient of $k$, which is $4$.

k=5b−34k = \frac{5b - 3}{4}

Conclusion

In this article, we have solved the linear equation $3 - 3k + 7k = 5b$ to find the value of the variable $k$. We have broken down the solution into three steps: combining like terms, isolating the variable, and solving for $k$. By following these steps, we have arrived at the solution:

k=5b−34k = \frac{5b - 3}{4}

Example Use Case

Suppose we are given the value of $b$ as $2$. We can substitute this value into the solution to find the value of $k$:

k=5(2)−34k = \frac{5(2) - 3}{4}

Simplifying the expression, we get:

k=10−34k = \frac{10 - 3}{4}

k=74k = \frac{7}{4}

Therefore, the value of $k$ is $\frac{7}{4}$.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS):

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

By following these steps and using algebraic techniques, you can solve linear equations with ease.

Common Mistakes to Avoid

When solving linear equations, it's easy to make mistakes. Here are some common mistakes to avoid:

  1. Not combining like terms: Make sure to combine like terms to simplify the equation.
  2. Not isolating the variable: Make sure to isolate the variable by getting all the terms with the variable on one side of the equation.
  3. Not checking the solution: Make sure to check the solution by plugging it back into the original equation.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

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 refresher on the concepts. In this article, we'll answer some frequently asked questions about solving linear equations, providing you with a deeper understanding of the subject.

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's an equation that can be written in the form of 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, follow these steps:

  1. Combine like terms.
  2. Isolate the variable by getting all the terms with the variable on one side of the equation.
  3. Solve for the variable.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when evaluating an expression. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate 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 combine like terms?

A: To combine like terms, look for terms that have the same variable raised to the same power. Then, add or subtract the coefficients of those terms.

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(s) is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable(s) is 2.

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

A: Yes, algebraic techniques are a powerful tool for solving linear equations. By using techniques such as combining like terms, isolating the variable, and solving for the variable, you can solve linear equations with ease.

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

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

  1. Not combining like terms.
  2. Not isolating the variable.
  3. Not checking the solution.

Q: How do I check my solution?

A: To check your solution, plug it back into the original equation and see if it's true. If it's not true, then your solution is incorrect.

Q: Can I use technology to solve linear equations?

A: Yes, technology can be a powerful tool for solving linear equations. Many graphing calculators and computer algebra systems can solve linear equations with ease.

Conclusion

Solving linear equations is a crucial skill for students to master. By following the steps outlined in this article and practicing regularly, you can become proficient in solving linear equations and tackle more complex mathematical problems with confidence. Remember to combine like terms, isolate the variable, and solve for the variable, and don't be afraid to ask for help if you need it.

Additional Resources

For more information on solving linear equations, check out the following resources:

  • Khan Academy: Solving Linear Equations
  • Mathway: Solving Linear Equations
  • Wolfram Alpha: Solving Linear Equations

Practice Problems

Try solving the following linear equations:

  1. 2x + 3 = 5
  2. x - 2 = 3
  3. 4x + 2 = 10

Answer Key

  1. x = 1
  2. x = 5
  3. x = 2