Solve For { X $} . . . { 8 - \frac{3}{8} X = -4 \}

by ADMIN 51 views

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 x. We will use the given equation 8 - (3/8)x = -4 as an example to demonstrate the step-by-step process of solving for x.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its components. The equation is in the form of a linear equation, which is typically written as ax + b = c, where a, b, and c are constants. In this case, the equation is 8 - (3/8)x = -4. Our goal is to isolate the variable x, which means we need to get x by itself on one side of the equation.

Step 1: Add (3/8)x to Both Sides

To isolate x, we need to get rid of the negative term (-3/8)x. We can do this by adding (3/8)x to both sides of the equation. This will cancel out the negative term, leaving us with a simpler equation.

8 - (3/8)x + (3/8)x = -4 + (3/8)x

Simplifying the equation, we get:

8 = -4 + (3/8)x

Step 2: Add 4 to Both Sides

Next, we need to get rid of the constant term -4 on the right-hand side of the equation. We can do this by adding 4 to both sides of the equation.

8 + 4 = -4 + 4 + (3/8)x

Simplifying the equation, we get:

12 = (3/8)x + 4

Step 3: Subtract 4 from Both Sides

Now, we need to isolate the term (3/8)x. We can do this by subtracting 4 from both sides of the equation.

12 - 4 = (3/8)x + 4 - 4

Simplifying the equation, we get:

8 = (3/8)x

Step 4: Multiply Both Sides by 8

Finally, we need to get rid of the fraction (3/8) by multiplying both sides of the equation by 8.

8 \* 8 = (3/8)x \* 8

Simplifying the equation, we get:

64 = 3x

Step 5: Divide Both Sides by 3

Now, we can finally isolate x by dividing both sides of the equation by 3.

64 / 3 = 3x / 3

Simplifying the equation, we get:

x = 64 / 3

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation. We used the equation 8 - (3/8)x = -4 as an example and showed how to isolate the variable x. By following these steps, you can solve any linear equation that involves isolating x. Remember to always add or subtract the same value to both sides of the equation and multiply or divide both sides by the same value to keep the equation balanced.

Tips and Tricks

  • Always check your work by plugging the solution back into the original equation.
  • Use a calculator to check your calculations and ensure that you are getting the correct answer.
  • Practice solving linear equations regularly to build your skills and confidence.

Common Mistakes

  • Forgetting to add or subtract the same value to both sides of the equation.
  • Multiplying or dividing both sides of the equation by the wrong value.
  • Not checking your work by plugging the solution back into the original equation.

Real-World Applications

Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects and the forces acting on them.
  • 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.

Conclusion

Introduction

In our previous article, we demonstrated the step-by-step process of solving a linear equation. However, we know that practice makes perfect, and sometimes, it's helpful to have a quick reference guide to answer common questions and clarify any doubts. In this article, we will provide a Q&A guide to help you better understand how to solve linear equations.

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 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?

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

  • The highest power of the variable (usually x) is 1.
  • The equation is in the form of ax + b = c, where a, b, and c are constants.
  • There are no squared or cubed 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 (usually x) is 1, whereas a quadratic equation is an equation in which the highest power of the variable (usually x) is 2.

Q: How do I solve a linear equation?

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

  1. Add or subtract the same value to both sides of the equation to isolate the variable.
  2. Multiply or divide both sides of the equation by the same value to eliminate any fractions.
  3. Check your work by plugging the solution back into the original equation.

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

A: When solving a linear equation, follow the order of operations:

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

Q: How do I handle fractions when solving a linear equation?

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

  1. Multiply or divide both sides of the equation by the denominator of the fraction to eliminate the fraction.
  2. Check your work by plugging the solution back into the original equation.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation with one variable, whereas a system of linear equations is a set of two or more linear equations with the same variable.

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

A: To solve a system of linear equations, follow these steps:

  1. Use the substitution method to substitute one equation into the other equation.
  2. Use the elimination method to eliminate one variable by adding or subtracting the equations.
  3. Check your work by plugging the solution back into both original equations.

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 add or subtract the same value to both sides of the equation.
  • Multiplying or dividing both sides of the equation by the wrong value.
  • Not checking your work by plugging the solution back into the original equation.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the step-by-step process outlined in this article and using the Q&A guide, you can better understand how to solve linear equations and avoid common mistakes. Remember to always check your work and practice regularly to build your skills and confidence.