Solve For F F F .${ \begin{array}{l} -9 + 8f = -6 + 5f \ f = \square \end{array} }$

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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 to find the value of ff. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.

The Linear Equation

The given linear equation is:

−9+8f=−6+5f-9 + 8f = -6 + 5f

Our goal is to isolate the variable ff and find its value.

Step 1: Add 9 to Both Sides

To start solving the equation, we need to get rid of the negative term on the left-hand side. We can do this by adding 9 to both sides of the equation.

−9+9+8f=−6+9+5f-9 + 9 + 8f = -6 + 9 + 5f

This simplifies to:

8f=3+5f8f = 3 + 5f

Step 2: Subtract 5f from Both Sides

Next, we need to get all the terms involving ff on one side of the equation. We can do this by subtracting 5f from both sides.

8f−5f=3+5f−5f8f - 5f = 3 + 5f - 5f

This simplifies to:

3f=33f = 3

Step 3: Divide Both Sides by 3

Now that we have the term involving ff isolated, we can solve for ff by dividing both sides of the equation by 3.

3f3=33\frac{3f}{3} = \frac{3}{3}

This simplifies to:

f=1f = 1

Conclusion

In this article, we solved a linear equation to find the value of ff. We broke down the solution into three steps, using clear and concise language to ensure that readers understand the process. By following these steps, we were able to isolate the variable ff and find its value.

Why is Solving Linear Equations Important?

Solving linear equations is an essential skill in mathematics, and it has numerous applications in real-life situations. For example, linear equations are used in physics to describe the motion of objects, in economics to model the behavior of markets, and in computer science to solve problems involving algorithms.

Tips for Solving Linear Equations

Here are some tips for solving linear equations:

  • Use inverse operations: To solve a linear equation, you need to use inverse operations to isolate the variable. For example, to solve the equation 2x+3=52x + 3 = 5, you need to subtract 3 from both sides and then divide both sides by 2.
  • Check your work: Once you have solved the equation, make sure to check your work by plugging the solution back into the original equation.
  • Use algebraic manipulations: Linear equations can be solved using algebraic manipulations such as adding, subtracting, multiplying, and dividing both sides of the equation.

Common Mistakes to Avoid

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

  • Not checking your work: Failing to check your work can lead to incorrect solutions.
  • Not using inverse operations: Failing to use inverse operations can lead to incorrect solutions.
  • Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.

Conclusion

Introduction

In our previous article, we solved a linear equation to find the value of ff. We broke down the solution into three steps and provided tips and common mistakes to avoid. 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(s) is 1. It can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable by using inverse operations. This involves adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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

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

  • Not checking your work
  • Not using inverse operations
  • Not simplifying the equation
  • Not following the order of operations (PEMDAS)

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

A: To check your work, plug the solution back into the original equation and verify that it is true. This will help you ensure that your solution is correct.

Q: What are some real-life applications of linear equations?

A: Linear equations have numerous real-life applications, including:

  • Physics: Linear equations are used to describe the motion of objects.
  • Economics: Linear equations are used to model the behavior of markets.
  • Computer Science: Linear equations are used to solve problems involving algorithms.
  • Engineering: Linear equations are used to design and optimize systems.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, combine like terms and eliminate any unnecessary variables. This will help you isolate the variable and solve the equation.

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, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, the equation 2x+3=52x + 3 = 5 is a linear equation, while the equation x2+4x+4=0x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to find the values of the variables that satisfy all the equations in the system. This can be done using substitution or elimination methods.

Conclusion

Solving linear equations is an essential skill in mathematics, and it has numerous applications in real-life situations. By following the steps outlined in this article and avoiding common mistakes, you can solve linear equations and find the value of the variable. Remember to check your work, simplify the equation, and use inverse operations to ensure that your solution is correct.

Additional Resources

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

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

Practice Problems

Try solving the following linear equations:

  1. 2x+3=52x + 3 = 5
  2. x−2=3x - 2 = 3
  3. 4x+2=104x + 2 = 10

Answer Key

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