Solve: 3.4x+4=1.6x−5

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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, namely the equation 3.4x + 4 = 1.6x - 5. We will break down the solution step by step, using a clear and concise approach that is easy to follow.

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. Linear equations can be written in the form ax + b = c, where a, b, and c are constants. In the equation 3.4x + 4 = 1.6x - 5, we have a = 3.4, b = 4, and c = -5.

The Goal of Solving a Linear Equation

The goal of solving a linear equation is to isolate the variable (in this case, x) on one side of the equation. This means that we want to get rid of the constants on the same side as the variable, leaving only the variable and its coefficient.

Step 1: Subtract 1.6x from Both Sides

To start solving the equation, we need to get all the x terms on one side of the equation. We can do this by subtracting 1.6x from both sides of the equation. This gives us:

3.4x + 4 - 1.6x = 1.6x - 5 - 1.6x

Simplifying the left-hand side, we get:

1.8x + 4 = -5

Step 2: Subtract 4 from Both Sides

Next, we need to get rid of the constant term on the same side as the variable. We can do this by subtracting 4 from both sides of the equation. This gives us:

1.8x + 4 - 4 = -5 - 4

Simplifying the left-hand side, we get:

1.8x = -9

Step 3: Divide Both Sides by 1.8

Finally, we need to isolate the variable by dividing both sides of the equation by its coefficient. In this case, the coefficient of x is 1.8. Dividing both sides by 1.8 gives us:

(1.8x) / 1.8 = (-9) / 1.8

Simplifying the left-hand side, we get:

x = -5

Conclusion

In this article, we solved the linear equation 3.4x + 4 = 1.6x - 5 using a step-by-step approach. We started by subtracting 1.6x from both sides of the equation, then subtracted 4 from both sides, and finally divided both sides by 1.8 to isolate the variable. The solution to the equation is x = -5.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you perform the operations in the correct order.
  • Make sure to simplify the equation at each step to avoid confusion.
  • If you're having trouble solving a linear equation, try using a different method or approach.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • 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 to Avoid

  • When solving linear equations, it's easy to make mistakes by not following the order of operations or not simplifying the equation at each step.
  • Make sure to check your work by plugging the solution back into the original equation to ensure that it's true.

Conclusion

Introduction

In our previous article, we solved the linear equation 3.4x + 4 = 1.6x - 5 using a step-by-step approach. In this article, we will answer some of the most 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 (in this case, x) is 1. Linear equations can be written in the form ax + b = c, where a, b, and c are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable (in this case, x) on one side of the equation. This means that you want to get rid of the constants on the same side as the variable, leaving only the variable and its coefficient.

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

A: When solving a linear equation, it's essential to follow the order of operations (PEMDAS) to ensure that you perform the operations in the correct order. PEMDAS stands for:

  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 simplify an equation?

A: To simplify an equation, you need to combine like terms and eliminate any unnecessary operations. For example, if you have the equation 2x + 3x = 5, you can simplify it by combining the like terms: 5x = 5.

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 (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, the equation x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Alternatively, you can try to factor the equation or use a graphing calculator to find the solutions.

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

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

  • Not following the order of operations (PEMDAS)
  • Not simplifying the equation at each step
  • Not checking your work by plugging the solution back into the original equation

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

A: To check your work when solving a linear equation, you need to plug the solution back into the original equation and verify that it's true. For example, if you solve the equation 2x + 3 = 5 and get x = 1, you can plug x = 1 back into the original equation to verify that it's true: 2(1) + 3 = 5.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following a step-by-step approach and using a clear and concise language, we can solve even the most complex linear equations. In this article, we answered some of the most frequently asked questions about solving linear equations, and we hope that this guide will help you to better understand how to solve linear equations.

Additional Resources

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

Practice Problems

  • Solve the equation 2x + 5 = 3x - 2
  • Solve the equation x^2 + 4x + 4 = 0
  • Solve the equation 3x - 2 = 2x + 1

Answer Key

  • Solve the equation 2x + 5 = 3x - 2: x = -3
  • Solve the equation x^2 + 4x + 4 = 0: x = -2
  • Solve the equation 3x - 2 = 2x + 1: x = 3