Solve $84 = 7x - 77$.

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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 linear equation, $84 = 7x - 77$, and provide a step-by-step guide on how to arrive at the solution.

Understanding the Equation

The given equation is a linear equation in one variable, x. It is in the form of $ax + b = c$, where a, b, and c are constants. In this case, a = 7, b = -77, and c = 84.

Step 1: Add 77 to Both Sides

To isolate the term with the variable, x, we need to get rid of the constant term, -77. We can do this by adding 77 to both sides of the equation.

84+77=7x−77+7784 + 77 = 7x - 77 + 77

This simplifies to:

161=7x161 = 7x

Step 2: Divide Both Sides by 7

Now that we have isolated the term with the variable, x, we need to get rid of the coefficient, 7. We can do this by dividing both sides of the equation by 7.

1617=7x7\frac{161}{7} = \frac{7x}{7}

This simplifies to:

23=x23 = x

Conclusion

In this article, we have solved the linear equation $84 = 7x - 77$ using a step-by-step approach. We added 77 to both sides of the equation to isolate the term with the variable, x, and then divided both sides by 7 to get the final solution.

Final Answer

The final answer to the equation $84 = 7x - 77$ is:

x=23x = 23

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
  • When adding or subtracting the same value to both sides of an equation, you are essentially "canceling out" that value.
  • When dividing both sides of an equation by a value, you are essentially "canceling out" that value.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
  • Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Common Mistakes

  • Not following the order of operations (PEMDAS) when solving linear equations.
  • Not isolating the term with the variable, x, before solving for x.
  • Not checking the solution to ensure that it satisfies the original equation.

Conclusion

Solving linear equations is a fundamental skill that has numerous real-world applications. By following a step-by-step approach and avoiding common mistakes, you can solve linear equations with confidence. In this article, we have solved the linear equation $84 = 7x - 77$ using a step-by-step approach, and we hope that you have learned something new and valuable.

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Introduction

In our previous article, we solved the linear equation $84 = 7x - 77$ using a step-by-step approach. In this article, we will answer some frequently asked questions about linear equations and provide additional tips and tricks for solving them.

Q: What is a linear equation?

A linear equation is an equation in which the highest power of the variable (x) is 1. It is typically written in the form of $ax + b = c$, where a, b, and c are constants.

Q: How do I know if an equation is linear?

To determine if an equation is linear, look for the following characteristics:

  • The highest power of the variable (x) is 1.
  • The equation is in the form of $ax + b = c$, where a, b, and c are constants.
  • The equation does not contain any exponents or roots.

Q: What is the difference between a linear equation and a quadratic equation?

A linear equation is an equation in which the highest power of the variable (x) is 1, while a quadratic equation is an equation in which the highest power of the variable (x) is 2. For example:

  • Linear equation: $2x + 3 = 5$
  • Quadratic equation: $x^2 + 4x + 4 = 0$

Q: How do I solve a linear equation?

To solve a linear equation, follow these steps:

  1. Add or subtract the same value to both sides of the equation to isolate the term with the variable (x).
  2. Multiply or divide both sides of the equation by the same value to eliminate the coefficient of the variable (x).
  3. Check the solution to ensure that it satisfies the original equation.

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

Some common mistakes to avoid when solving linear equations include:

  • Not following the order of operations (PEMDAS) when solving the equation.
  • Not isolating the term with the variable (x) before solving for x.
  • Not checking the solution to ensure that it satisfies the original equation.

Q: How do I check if a solution satisfies the original equation?

To check if a solution satisfies the original equation, substitute the solution into the equation and simplify. If the resulting equation is true, then the solution is correct.

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

Linear equations have numerous real-world applications, including:

  • Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
  • Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Q: Can linear equations be used to model non-linear phenomena?

While linear equations are typically used to model linear phenomena, they can also be used to model non-linear phenomena by using techniques such as linearization or approximation.

Q: What are some advanced topics in linear equations?

Some advanced topics in linear equations include:

  • Systems of linear equations
  • Matrices and determinants
  • Linear transformations and eigenvectors
  • Linear programming and optimization

Conclusion

In this article, we have answered some frequently asked questions about linear equations and provided additional tips and tricks for solving them. We hope that you have learned something new and valuable, and that you will be able to apply this knowledge to real-world problems.