Solve For { X $} . . . { 2x - 34 = -20 \}

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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, which is a first-degree equation in one variable. We will use the given equation 2x - 34 = -20 as an example to demonstrate the step-by-step process of solving linear equations.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants. Linear equations can be solved using various methods, including algebraic manipulation, graphical representation, and numerical methods.

The Given Equation

The given equation is 2x - 34 = -20. This equation is a linear equation in one variable, x. Our goal is to solve for x, which means we need to isolate x on one side of the equation.

Step 1: Add 34 to Both Sides

To solve for x, we need to get rid of the constant term (-34) on the left side of the equation. We can do this by adding 34 to both sides of the equation. This will keep the equation balanced and allow us to isolate x.

2x - 34 + 34 = -20 + 34

This simplifies to:

2x = 14

Step 2: Divide Both Sides by 2

Now that we have 2x on the left side of the equation, we need to get rid of the coefficient (2) by dividing both sides of the equation by 2. This will give us the value of x.

(2x) / 2 = 14 / 2

This simplifies to:

x = 7

Conclusion

In this article, we solved a linear equation using the step-by-step process of adding and dividing both sides of the equation. We started with the given equation 2x - 34 = -20 and isolated x by adding 34 to both sides and then dividing both sides by 2. The final solution is x = 7.

Tips and Tricks

  • When solving linear equations, always check your work by plugging the solution back into the original equation.
  • Use the order of operations (PEMDAS) to simplify the equation and avoid errors.
  • If you are given a linear equation with a variable on both sides, try to isolate one variable by adding or subtracting the same value to both sides.

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

  • Not checking your work by plugging the solution back into the original equation.
  • Not using the order of operations (PEMDAS) to simplify the equation.
  • Not isolating one variable by adding or subtracting the same value to both sides.

Conclusion

Introduction

In our previous article, we discussed the step-by-step process of solving linear equations. However, we understand that sometimes, it's not enough to just follow a set of instructions. You may have questions, doubts, or concerns that need to be addressed. That's why we've put together this Q&A guide to help you better understand linear equations and how to solve them.

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. It can be written in the form 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 (x) is 1.
  • The equation can be written in the form ax + b = c.
  • The equation does not contain any squared or higher powers of the variable.

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 (x) is 1, whereas 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 with a variable on both sides?

A: To solve a linear equation with a variable on both sides, try to isolate one variable by adding or subtracting the same value to both sides. For example:

  • Equation: x + 2 = 3x - 1
  • Solution: Subtract x from both sides: 2 = 2x - 1
  • Add 1 to both sides: 3 = 2x
  • Divide both sides by 2: x = 3/2

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:

  • P: Parentheses (evaluate expressions inside parentheses first)
  • E: Exponents (evaluate any exponential expressions next)
  • M: Multiplication and Division (evaluate any multiplication and division operations from left to right)
  • A: Addition and Subtraction (finally, evaluate any addition and subtraction operations from left to right)

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

A: To check your work, plug the solution back into the original equation and simplify. If the equation is true, then your solution is correct. For example:

  • Equation: 2x - 3 = 5
  • Solution: x = 4
  • Check: 2(4) - 3 = 5
  • Simplify: 8 - 3 = 5
  • True!

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

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

  • Not checking your work by plugging the solution back into the original equation.
  • Not using the order of operations (PEMDAS) to simplify the equation.
  • Not isolating one variable by adding or subtracting the same value to both sides.

Conclusion

We hope this Q&A guide has helped you better understand linear equations and how to solve them. Remember to check your work, use the order of operations, and isolate one variable to avoid common mistakes. If you have any further questions or concerns, feel free to ask!