Solve The Equation And Check Your Proposed Solution.$\[ X + \frac{1}{7} = -\frac{1}{7} \\]\[$ X = [ \square \$\]

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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 a specific type of linear equation, which involves isolating the variable on one side of the equation. We will use a step-by-step approach to solve the equation and check our proposed solution.

The Equation

The given equation is:

x+17=−17{ x + \frac{1}{7} = -\frac{1}{7} }

This is a linear equation in one variable, where the variable is x. The equation involves a fraction, which can be challenging to work with. However, with the right approach, we can simplify the equation and solve for x.

Step 1: Subtract the Fraction from Both Sides

To isolate the variable x, we need to get rid of the fraction on the left-hand side of the equation. We can do this by subtracting the fraction from both sides of the equation. This will give us:

x=−17−17{ x = -\frac{1}{7} - \frac{1}{7} }

Step 2: Simplify the Right-Hand Side

Now that we have subtracted the fraction from both sides, we can simplify the right-hand side of the equation. To do this, we need to find a common denominator for the two fractions. In this case, the common denominator is 7. So, we can rewrite the equation as:

x=−27{ x = -\frac{2}{7} }

Step 3: Check the Proposed Solution

Now that we have solved the equation, we need to check our proposed solution to make sure it is correct. To do this, we can substitute the value of x back into the original equation and see if it is true. If it is true, then our proposed solution is correct.

Substituting x = -2/7 into the original equation, we get:

−27+17=−17{ -\frac{2}{7} + \frac{1}{7} = -\frac{1}{7} }

Simplifying the left-hand side of the equation, we get:

−17=−17{ -\frac{1}{7} = -\frac{1}{7} }

This is true, so our proposed solution is correct.

Conclusion

In this article, we solved a linear equation involving a fraction and checked our proposed solution. We used a step-by-step approach to isolate the variable x and simplify the right-hand side of the equation. We then checked our proposed solution by substituting the value of x back into the original equation. This ensured that our solution was correct and provided a clear understanding of the problem.

Tips and Tricks

  • When working with fractions, it's essential to find a common denominator to simplify the equation.
  • When subtracting fractions, make sure to subtract the numerators and keep the denominator the same.
  • When checking the proposed solution, substitute the value of x back into the original equation to ensure it is true.

Real-World Applications

Linear equations are used in various real-world applications, such as:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Common Mistakes

  • Forgetting to find a common denominator when working with fractions.
  • Subtracting the denominator instead of the numerator when subtracting fractions.
  • Not checking the proposed solution to ensure it is true.

Practice Problems

Try solving the following linear equations:

  1. x+3=7{ x + 3 = 7 }
  2. x−2=5{ x - 2 = 5 }
  3. x+12=−12{ x + \frac{1}{2} = -\frac{1}{2} }

Remember to follow the steps outlined in this article to solve the equations and check your proposed solutions.

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Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. It is a simple equation that can be solved using basic algebraic operations.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable (usually x) on one side of the equation.
  3. Use basic algebraic operations (addition, subtraction, multiplication, and division) to solve for x.

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:

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

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

A: When working with fractions in a linear equation, follow these steps:

  1. Find a common denominator for the fractions.
  2. Add or subtract the fractions by adding or subtracting the numerators and keeping the denominator the same.
  3. Simplify the equation by combining like terms.

Q: What is the order of operations when solving a linear equation?

A: The order of operations when solving a linear equation 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 check my proposed solution to a linear equation?

A: To check your proposed solution to a linear equation, substitute the value of x back into the original equation and see if it is true. If it is true, then your proposed solution is correct.

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

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

  • Forgetting to find a common denominator when working with fractions.
  • Subtracting the denominator instead of the numerator when subtracting fractions.
  • Not checking the proposed solution to ensure it is true.

Q: Can linear equations be used in real-world applications?

A: Yes, linear equations are used in various real-world applications, such as:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by:

  • Working through example problems in a textbook or online resource.
  • Creating your own practice problems and solving them.
  • Using online resources, such as Khan Academy or Mathway, to practice solving linear equations.

Q: What are some advanced topics in linear equations?

A: Some advanced topics in linear equations include:

  • Systems of linear equations: Solving multiple linear equations simultaneously.
  • Linear inequalities: Solving linear equations with inequalities.
  • Matrix algebra: Using matrices to solve linear equations.

Q: Can I use technology to solve linear equations?

A: Yes, you can use technology to solve linear equations, such as:

  • Graphing calculators: Using a graphing calculator to visualize the solution to a linear equation.
  • Online resources: Using online resources, such as Khan Academy or Mathway, to solve linear equations.
  • Computer algebra systems: Using computer algebra systems, such as Mathematica or Maple, to solve linear equations.