Solve For { X$}$ In The Equation:${ -1 = \frac{5+x}{6} }$

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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 type of linear equation, where the variable x is isolated on one side of the equation. We will use the equation -1 = (5+x)/6 as a case study to demonstrate the step-by-step process of solving for x.

Understanding the Equation

Before we dive into the solution, let's take a closer look at the equation -1 = (5+x)/6. This equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, x) is 1. The equation is also a rational equation, which means it contains fractions.

Step 1: Multiply Both Sides by the Denominator

To solve for x, we need to isolate the variable on one side of the equation. The first step is to multiply both sides of the equation by the denominator, which is 6. This will eliminate the fraction and make it easier to work with.

-1 = (5+x)/6
-1 × 6 = (5+x)/6 × 6
-6 = 5 + x

Step 2: Subtract 5 from Both Sides

Now that we have -6 = 5 + x, we can subtract 5 from both sides to isolate the variable x.

-6 = 5 + x
-6 - 5 = 5 - 5 + x
-11 = x

Step 3: Simplify the Equation

The final step is to simplify the equation by combining like terms. In this case, there are no like terms to combine, so the equation is already simplified.

Conclusion

In this article, we solved the linear equation -1 = (5+x)/6 by multiplying both sides by the denominator, subtracting 5 from both sides, and simplifying the equation. The final solution is x = -11.

Real-World Applications

Solving linear equations is a crucial skill in many real-world applications, including:

  • Science and Engineering: Linear equations are used to model real-world phenomena, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
  • Economics: Linear equations are used to model economic systems, including supply and demand curves, and the behavior of financial markets.
  • Computer Science: Linear equations are used in computer science to solve problems in computer graphics, game development, and machine learning.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

  • Use the distributive property: The distributive property states that a(b+c) = ab + ac. This property can be used to simplify linear equations by distributing the coefficients to the variables.
  • Use inverse operations: Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division. Using inverse operations can help you isolate the variable on one side of the equation.
  • Check your work: It's always a good idea to check your work by plugging the solution back into the original equation. This will help you ensure that your solution is correct.

Common Mistakes

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

  • Forgetting to multiply both sides by the denominator: When multiplying both sides by the denominator, make sure to multiply both sides by the same value.
  • Forgetting to simplify the equation: Make sure to simplify the equation by combining like terms.
  • 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 know that practice makes perfect, and there's no better way to practice than by answering questions. In this article, we'll provide a Q&A guide to help you reinforce your understanding of solving linear equations.

Q1: What is a linear equation?

A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. For example, 2x + 3 = 5 is a linear equation.

Q2: How do I solve a linear equation?

To solve a linear equation, follow these steps:

  1. Multiply both sides by the denominator (if the equation contains fractions).
  2. Use inverse operations to isolate the variable on one side of the equation.
  3. Simplify the equation by combining like terms.
  4. Check your work by plugging the solution back into the original equation.

Q3: What is the distributive property?

The distributive property states that a(b+c) = ab + ac. This property can be used to simplify linear equations by distributing the coefficients to the variables.

Q4: How do I use inverse operations to solve a linear equation?

Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division. To use inverse operations, follow these steps:

  1. Identify the inverse operation of the operation on the variable.
  2. Apply the inverse operation to both sides of the equation.
  3. Simplify the equation by combining like terms.

Q5: 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 (in this case, x) is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, x^2 + 3x + 2 = 0 is a quadratic equation.

Q6: How do I check my work when solving a linear equation?

To check your work, plug the solution back into the original equation. If the solution satisfies the equation, then it is correct. If not, then you need to re-evaluate your solution.

Q7: What are some common mistakes to avoid when solving linear equations?

Some common mistakes to avoid when solving linear equations include:

  • Forgetting to multiply both sides by the denominator.
  • Forgetting to simplify the equation.
  • Not checking your work.

Q8: How do I apply linear equations to real-world problems?

Linear equations can be applied to a wide range of real-world problems, including:

  • Science and engineering: Linear equations are used to model real-world phenomena, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
  • Economics: Linear equations are used to model economic systems, including supply and demand curves, and the behavior of financial markets.
  • Computer science: Linear equations are used in computer science to solve problems in computer graphics, game development, and machine learning.

Q9: What are some tips and tricks for solving linear equations?

Some tips and tricks for solving linear equations include:

  • Using the distributive property to simplify the equation.
  • Using inverse operations to isolate the variable on one side of the equation.
  • Checking your work by plugging the solution back into the original equation.

Q10: How do I practice solving linear equations?

To practice solving linear equations, try the following:

  • Work through practice problems in a textbook or online resource.
  • Use online tools or apps to generate random linear equations and solve them.
  • Join a study group or find a study partner to work through problems together.

Conclusion

Solving linear equations is a crucial skill in mathematics, and it has many real-world applications. By following the step-by-step process outlined in this article, you can solve linear equations with ease. Remember to use the distributive property, inverse operations, and check your work to ensure that your solution is correct. With practice and patience, you will become proficient in solving linear equations and be able to apply this skill to a wide range of problems.